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Question 1:

Find the zeros of the quadratic polynomial (x2 + 3x − 10) and verify the relation between its zeros and coefficients.

Question 2:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
${x}^{2}-2x-8$

${x}^{2}-2x-8=0\phantom{\rule{0ex}{0ex}}⇒{x}^{2}-4x+2x-8=0\phantom{\rule{0ex}{0ex}}⇒x\left(x-4\right)+2\left(x-4\right)=0$

Sum of zeroes =  4+(3)=71=(coefficient of x)(coefficient of x2)
Product of zeroes =  (4)(3)=121=constant termcoefficient of x2

Question 3:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
${x}^{2}+7x+12$

${x}^{2}+7x+12=0\phantom{\rule{0ex}{0ex}}⇒{x}^{2}+4x+3x+12=0\phantom{\rule{0ex}{0ex}}⇒x\left(x+4\right)+3\left(x+4\right)=0$

Sum of zeroes =
Product of zeroes =

Question 4:

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:
2x2 x – 6

Let f(x) = 2x x – 6

Hence, all the zeroes of the polynomial f(x) are .

Now,

Hence, the relationship between the zeros and the coefficients is verified.

Question 5:

Find the zeros of the quadratic polynomial 4x2 − 4x − 3 and verify the relation between the zeros and the coefficients.

Question 6:

Find the zeros of the quadratic polynomial 5x2 − 4 − 8x and verify the relationship between the zeros and the coefficients of the given polynomial.

Question 7:

Find the zeros of the quadratic polynomial 2x2 − 11x + 15 and verify the relation between the zeros and the coefficients.

Question 8:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
$2\sqrt{3}{x}^{2}-5x+\sqrt{3}$

$2\sqrt{3}{x}^{2}-5x+\sqrt{3}\phantom{\rule{0ex}{0ex}}⇒2\sqrt{3}{x}^{2}-2x-3x+\sqrt{3}\phantom{\rule{0ex}{0ex}}⇒2x\left(\sqrt{3}x-1\right)-\sqrt{3}\left(\sqrt{3}x-1\right)=0$

Sum of zeroes =  4+(3)=71=(coefficient of x)(coefficient of x2)
Product of zeroes =  (4)(3)=121=constant termcoefficient of x2

Question 9:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
$4{x}^{2}-4x+1$

Sum of zeroes =  4+(3)=71=(coefficient of x)(coefficient of x2)
Product of zeroes =  (4)(3)=121=constant termcoefficient of x2

Question 10:

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:
3x2 x – 4

Let f(x) = 3x x – 4

Hence, all the zeroes of the polynomial f(x) are .

Now,

Hence, the relationship between the zeros and the coefficients is verified.

Question 11:

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:
5x2 + 10x

Let f(x) = 5x+ 10x

Hence, all the zeroes of the polynomial f(x) are .

Now,

Hence, the relationship between the zeros and the coefficients is verified.

Question 12:

Find the zeros of the quadratic polynomial (8 x2 − 4) and verify the relation between the zeros and the coefficients.

Question 13:

If α and β are the zeros of the polynomial $p\left(x\right)=2{x}^{2}+5x+k$ satisfying the relation ${\mathrm{\alpha }}^{2}+{\mathrm{\beta }}^{2}+\mathrm{\alpha \beta }=\frac{21}{4}$ then find the value of k.

Let α and β be the zeroes of the polynomial $p\left(x\right)=2{x}^{2}+5x+k$.

Now, using (1)

Hence, the value of k is 2.

Question 14:

Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial.

Question 15:

Find the quadratic polynomial, the sum of whose zeros is 0 and their product is −1. Hence, find the zeros of the polynomial.

Question 16:

Find the quadratic polynomial, the sum of whose zeros is $\left(\frac{5}{2}\right)$ and their product is 1. Hence, find the zeros of the polynomial.

Question 17:

Find the quadratic polynomial whose zeros are 2 and −6. Verify the relation between the coefficients and the zeros of the polynomial.

Question 18:

Find the quadratic polynomial whose zeros are $\frac{2}{3}$ and $\frac{-1}{4}$. Verify the relation between the coefficients and the zeros of the polynomial.

Question 19:

If $\left(x+a\right)$ is a factor of the polynomial $2{x}^{2}+2ax+5x+10$, find the value of a.

Given: $\left(x+a\right)$ is a factor of $2{x}^{2}+2ax+5x+10$
So, we have
$x+a=0\phantom{\rule{0ex}{0ex}}⇒x=-a$
Now, It will satisfy  the above polynomial.
Therefore, we will get
$2{\left(-a\right)}^{2}+2a\left(-a\right)+5\left(-a\right)+10=0\phantom{\rule{0ex}{0ex}}⇒2{a}^{2}-2{a}^{2}-5a+10=0\phantom{\rule{0ex}{0ex}}⇒-5a=-10\phantom{\rule{0ex}{0ex}}⇒a=2$

Question 20:

If $x=\frac{2}{3}$ and $x=-3$ are the roots of the quadratic equation $a{x}^{2}+7x+b=0$ then find the values of a and b.

Given: $a{x}^{2}+7x+b=0$
Since, $x=\frac{2}{3}$ is the root of the above quadratic equation
Hence, It will satisfy the above equation.
Therefore, we will get

Since, $x=-3$ is the root of the above quadratic equation
Hence, It will satisfy the above equation.
Therefore, we will get

From (1) and (2), we get

Question 18:

Obtain all other zeros of (x4 + 4x3 − 2x2 − 20x − 15) if two of its zeros are $\sqrt{5}$ and $-\sqrt{5}$.

Question 23:

If α and β are the zeroes of a polynomial f(x) = 5x2 − 7x +1, find the value of $\left(\frac{1}{\alpha }+\frac{1}{\beta }\right)$

By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes =  and Product of zeroes =

Question 24:

If α and β are the zeroes of a polynomial f(x) = x2 + x − 2, find the value of $\left(\frac{1}{\alpha }-\frac{1}{\beta }\right)$

By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =

$\because {\left(\frac{1}{\alpha }-\frac{1}{\beta }\right)}^{2}=\frac{9}{4}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{\alpha }-\frac{1}{\beta }=±\frac{3}{2}$

Question 25:

If the zeros of the polynomial f(x) = x3 − 3x2 + x + 1 are (a − b), a and (a + b), Find a and b.

By using the relationship between the zeroes of the cubic ploynomial.
We have, Sum of zeroes =
$\therefore a-b+a+a+b=\frac{-\left(-3\right)}{1}\phantom{\rule{0ex}{0ex}}⇒3a=3\phantom{\rule{0ex}{0ex}}⇒a=1$

Now, Product of zeros =

Question 1:

Verity that 3, −2, 1 are the zeros of the cubic polynomial p(x) = x3 − 2x2 − 5x + 6 and verify the relation between its zeros and coefficients.

Question 2:

Verify that 5, −2 and $\frac{1}{3}$ are the zeros of the cubic polynomial p(x) = 3x3 − 10x2 − 27x + 10 and verify the relation between its zeros and coefficients.

Question 3:

Find a cubic polynomial whose zeroes are 2, −3 and 4

If the zeroes of the cubic polynomial are a, b and c then the cubic polynomial can be found as
${x}^{3}-\left(a+b+c\right){x}^{2}+\left(ab+bc+ca\right)x-abc$                                  ...(1)
Let
Substituting the values in (1), we get
${x}^{3}-\left(2-3+4\right){x}^{2}+\left(-6-12+8\right)x-\left(-24\right)\phantom{\rule{0ex}{0ex}}⇒{x}^{3}-3{x}^{2}-10x+24$

Question 4:

Find a cubic polynomial whose zeroes are $\frac{1}{2}$, 1 and −3.

If the zeroes of the cubic polynomial are a, b and c then the cubic polynomial can be found as
${x}^{3}-\left(a+b+c\right){x}^{2}+\left(ab+bc+ca\right)x-abc$                                ...(1)
Let
Substituting the values in (1), we get
${x}^{3}-\left(\frac{1}{2}+1-3\right){x}^{2}+\left(\frac{1}{2}-3-\frac{3}{2}\right)x-\left(\frac{-3}{2}\right)\phantom{\rule{0ex}{0ex}}⇒{x}^{3}-\left(\frac{-3}{2}\right){x}^{2}-4x+\frac{3}{2}\phantom{\rule{0ex}{0ex}}⇒2{x}^{3}+3{x}^{2}-8x+3$

Question 5:

Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time and the product of its zeroes are 5, −2 and −24 respectively.

We know the sum, sum of the product of the zeroes taken two at a time and the product of the zeroes of a cubic polynomial then the cubic polynomial can be found as
x3 −(Sum of the zeroes)x2 + (sum of the product of the zeroes taking two at a time)x − Product of zeroes
Therefore, the required polynomial is
${x}^{3}-5{x}^{2}-2x+24$

Question 6:

Find the quotient and the remainder when:
$f\left(x\right)={x}^{3}-3{x}^{2}+5x-3$ is divided by $g\left(x\right)={x}^{2}-2$

Quotient  $q\left(x\right)=x-3$
Remainder  $r\left(x\right)=7x-9$

Question 7:

Find the quotient and remainder when:
$f\left(x\right)={x}^{4}-3{x}^{2}+4x+5$ is divided by $g\left(x\right)={x}^{2}+1-x$

Quotient  $q\left(x\right)={x}^{2}+x-3$
Remainder  $r\left(x\right)=8$

Question 8:

Find the quotient and remainder when
$f\left(x\right)={x}^{4}-5x+6$ is divided by $g\left(x\right)=2-{x}^{2}$

We can write

and

Quotient  $q\left(x\right)=-{x}^{2}-2$
Remainder  $r\left(x\right)=-5x+10$

Question 9:

By actual division, show that ${x}^{2}-3$ is a factor of $2{x}^{4}+3{x}^{3}-2{x}^{2}-9x-12$

Let $f\left(x\right)=2{x}^{4}+3{x}^{3}-2{x}^{2}-9x-12$ and $g\left(x\right)={x}^{2}-3$

Quotient  $q\left(x\right)=2{x}^{2}+3x+4$
Remainder  $r\left(x\right)=0$
Since, the remainder is 0.
Hence, ${x}^{2}-3$ is a factor of $2{x}^{4}+3{x}^{3}-2{x}^{2}-9x-12$

Question 10:

If the polynomial (x4 + 2x3 + 8x2 + 12x + 18) is divided by another polynomial (x2 + 5), the remainder comes out to be (px + q). Find the values of p and q.

Let f(x) = x4 + 2x+ 8x+ 12x + 18

It is given that when f(x) is divisible by x2 + 5, the remainder comes out to be px + q.

On division, we get the quotient x2 + 2x + 3 and the remainder 2x + 3.

Since, the remainder comes out to be px + q.

Therefore, p = 2 and q = 3.

Hence, the values of p and q are 2 and 3 respectively.

Question 11:

On dividing, $3{x}^{3}+{x}^{2}+2x+5$ by a polynomial g(x), the quotient and remainder are $3x-5$ and $9x+10$ respectively. Find g(x)

By using division rule, we have
Divided = Quotient × Divisor + Remainder
$\therefore 3{x}^{3}+{x}^{2}+2x+5=\left(3x-5\right)g\left(x\right)+9x+10\phantom{\rule{0ex}{0ex}}⇒3{x}^{3}+{x}^{2}+2x+5-9x-10=\left(3x-5\right)g\left(x\right)\phantom{\rule{0ex}{0ex}}⇒3{x}^{3}+{x}^{2}-7x-5=\left(3x-5\right)g\left(x\right)\phantom{\rule{0ex}{0ex}}⇒g\left(x\right)=\frac{3{x}^{3}+{x}^{2}-7x-5}{3x-5}$

$\therefore g\left(x\right)={x}^{2}+2x+1$

Question 12:

Verify division algorithm for the polynomials $f\left(x\right)=8+20x+{x}^{2}-6{x}^{3}$ and $g\left(x\right)=2+5x-3{x}^{2}$

We can write and

Quotient = $2x+3$
Remainder = $x+2$

By using division rule, we have

Divided = Quotient × Divisor + Remainder

Question 13:

It is given that −1 is one of the zeros of the polynomial x3 + 2x2 − 11x − 12. Find all the given zeros of the given polynomial.

Question 14:

If 1 and −2 are two zeros of the polynomial (x3 − 4x2 − 7x + 10), find its third zero.

Question 15:

If 3 and −3 are two zeros of the polynomial (x4 + x3 − 11x2 − 9x + 18), find all the zeros of the given polynomial.

Question 16:

If 2 and –2 are two zeros of the polynomial 2x4 – 5x3 – 11x2  + 20x + 12, find all the zeros of the given polynomial.

Let f(x) = 2x4 – 5x3 – 11x2  + 20x + 12

It is given that 2 and –2 are two zeroes of f(x)

Thus,  f(x) is completely divisible by (x + 2) and (x – 2).

Therefore, one factor of f(x) is (x2 – 4).

We get another factor of f(x) by dividing it with (x2 – 4).

On division, we get the quotient 2x2 – 5x  – 3.

Hence, all the zeroes of the polynomial f(x) are

Question 17:

Find all the zeros of (x4 + x3 − 23x2 − 3x + 60), if it is given that two of its zeros are $\sqrt{3}$ and $-\sqrt{3}$.

Question 19:

Obtain all the zeros of the polynomial x4 + x3 – 14x2 – 2x + 24 if two of its zeros are .

Let f(x) = xx– 14x– 2x + 24

It is given that  are two zeroes of f(x)

Thus,  f(x) is completely divisible by (x + $\sqrt{2}$) and (x – $\sqrt{2}$).

Therefore, one factor of f(x) is (x2 – 2).

We get another factor of f(x) by dividing it with (x2 – 2).

On division, we get the quotient x2 + x  – 12.

Hence, all the zeroes of the polynomial f(x) are

Question 20:

Find all the zeros of 2x4 – 13x3 + 19x2 + 7x – 3 if two of its zeros are .

Let f(x) = 2x– 13x+ 19x+ 7x – 3

It is given that  are two zeroes of f(x)

Thus,  f(x) is completely divisible by (x $-2-\sqrt{3}$) and (x – $2+\sqrt{3}$).

Therefore, one factor of f(x) is $\left({\left(x-2\right)}^{2}-3\right)$
$⇒$one factor of f(x) is (x2 – 4x + 1)

We get another factor of f(x) by dividing it with (x2 – 4x + 1).

On division, we get the quotient 2x2 – 5x  – 3.

Hence, all the zeroes of the polynomial f(x) are

Question 21:

One zero of the polynomial . Find the other zeros of the polynomial.

Let f(x) = 3x3 + 16x2 + 15x – 18

It is given that one of its zeroes is $\frac{2}{3}$.

Therefore, one factor of f(x) is (x – $\frac{2}{3}$).

We get another factor of f(x) by dividing it with (x – $\frac{2}{3}$).

On division, we get the quotient 3x+ 18x + 27.

Hence, other zero of the polynomial f(x) is –3.

Question 22:

Find all the zeros of 2x4 – 3x3 – 3x2 + 6x – 2 if it is given that two of its zeros are 1 and $\frac{1}{2}$.

Let f(x) = 2x– 3x– 3x+ 6x – 2

It is given that 1 and $\frac{1}{2}$ are two zeroes of f(x).

Thus,  f(x) is completely divisible by (x – 1) and (x – $\frac{1}{2}$).

Therefore, one factor of f(x) is $\left(x-1\right)\left(x-\frac{1}{2}\right)$
$⇒$one factor of f(x) is $\left({x}^{2}-\frac{3}{2}x+\frac{1}{2}\right)$

We get another factor of f(x) by dividing it with $\left({x}^{2}-\frac{3}{2}x+\frac{1}{2}\right)$.

On division, we get the quotient 2x2 – 4.

Hence, all the zeroes of the polynomial f(x) are

Question 23:

Find all the zeros of the polynomial (2x4 − 11x3 + 7x2 + 13x), it being given that two if its zeros are $3+\sqrt{2}$ and $3-\sqrt{2}$.

Question 1:

If one zero of the polynomial ${x}^{2}-4x+1$  is $2+\sqrt{3}$ . Write the other zero.  [CBSE 2010]

Let the other zeroes of ${x}^{2}-4x+1$ be a.
By using the relationship between the zeroes of the quadratic ploynomial.
We have, Sum of zeroes =
$\therefore 2+\sqrt{3}+a=\frac{-\left(-4\right)}{1}\phantom{\rule{0ex}{0ex}}⇒a=2-\sqrt{3}$
Hence, the other zeroes of ${x}^{2}-4x+1$ is $2-\sqrt{3}$.

Question 2:

Find the zeros of the polynomial x2 + x − p(p + 1).   [CBSE 2011]

$f\left(x\right)={x}^{2}+x-p\left(p+1\right)$
By adding and subtracting px, we get
$f\left(x\right)={x}^{2}+px+x-px-p\left(p+1\right)\phantom{\rule{0ex}{0ex}}={x}^{2}+\left(p+1\right)x-px-p\left(p+1\right)\phantom{\rule{0ex}{0ex}}=x\left[x+\left(p+1\right)\right]-p\left[x+\left(p+1\right)\right]$

So, the zeros of f(x) are −(p + 1) and p.

Question 3:

Find the zeros of the polynomial x2 − 3x − m(m + 3).   [CBSE 2011]

$f\left(x\right)={x}^{2}-3x-m\left(m+3\right)$
By adding and subtracting mx, we get
$f\left(x\right)={x}^{2}-mx-3x+mx-m\left(m+3\right)\phantom{\rule{0ex}{0ex}}=x\left[x-\left(m+3\right)\right]+m\left[x-\left(m+3\right)\right]\phantom{\rule{0ex}{0ex}}=\left[x-\left(m+3\right)\right]\left(x+m\right)$

So, the zeros of f(x) are −m and m + 3.

Question 4:

If α, β are the zeros of a polynomial such that α + β = 6 and αβ = 4 the write the polynomial.   [CBSE 2010]

If the zeroes of the quadratic polynomial are α and β then the quadratic polynomial can be found as
x2 − (α + β)x + αβ                .....(1)
Substituting the values in (1), we get
x2 − 6x + 4

Question 5:

If one zero of the quadratic polynomial kx2 + 3x + k is 2 then find the value of k.

Given: x = 2 is one zero of the quadratic polynomial kx2 + 3x + k
Therefore, It will satisfy the above polynomial.
Now, we have
$k{\left(2\right)}^{2}+3\left(2\right)+k=0\phantom{\rule{0ex}{0ex}}⇒4k+6+k=0\phantom{\rule{0ex}{0ex}}⇒5k+6=0\phantom{\rule{0ex}{0ex}}⇒k=-\frac{6}{5}$

Question 6:

If 3 is a zero of the polynomial 2x2 + x + k, find the value of k.  [CBSE 2010]

Given: x = 3 is one zero of the polynomial 2x2 + x + k
Therefore, It will satisfy the above polynomial.
Now, we have
$2{\left(3\right)}^{2}+3+k=0\phantom{\rule{0ex}{0ex}}⇒21+k=0\phantom{\rule{0ex}{0ex}}⇒k=-21$

Question 7:

If −4 is a zero of the quadratic polynomial x2x − (2k + 2) then find the value of k.

Given: x = −4 is one zero of the polynomial x2x −(2k + 2)
Therefore, It will satisfy the above polynomial.
Now, we have
${\left(-4\right)}^{2}-\left(-4\right)-\left(2k+2\right)=0\phantom{\rule{0ex}{0ex}}⇒16+4-2k-2=0\phantom{\rule{0ex}{0ex}}⇒-2k=-18\phantom{\rule{0ex}{0ex}}⇒k=9$

Question 8:

If 1 is a zero of the polynomial ax2 − 3(a − 1) x − 1, then find the value of a.

Given: x = 1 is one zero of the polynomial ax2 − 3(a − 1) x − 1
Therefore, It will satisfy the above polynomial.
Now, we have
$a{\left(1\right)}^{2}-3\left(a-1\right)1-1=0\phantom{\rule{0ex}{0ex}}⇒a-3a+3-1=0\phantom{\rule{0ex}{0ex}}⇒-2a=-2\phantom{\rule{0ex}{0ex}}⇒a=1$

Question 9:

If −2 is a zero of the polynomial 3x2 + 4x + 2k then find the value of k.

Given: x = −2 is one zero of the polynomial 3x2 + 4x + 2k
Therefore, It will satisfy the above polynomial.
Now, we have
$3{\left(-2\right)}^{2}+4\left(-2\right)+2k=0\phantom{\rule{0ex}{0ex}}⇒12-8+2k=0\phantom{\rule{0ex}{0ex}}⇒k=-2$

Question 10:

Write the zeros of the polynomial x2 −− 6

$f\left(x\right)={x}^{2}-x-6\phantom{\rule{0ex}{0ex}}={x}^{2}-3x+2x-6\phantom{\rule{0ex}{0ex}}=x\left(x-3\right)+2\left(x-3\right)$

So, the zeros of f(x) are 3 and −2.

Question 11:

If the sum of the zeros of the quadratic polynomial kx2 − 3x + 5 is 1, write the value of k.

By using the relationship between the zeros of the quadratic ploynomial.
We have
Sum of zeroes =
$⇒1=\frac{-\left(-3\right)}{k}\phantom{\rule{0ex}{0ex}}⇒k=3$

Question 12:

If the product of the zeros of the quadratic polynomial x2 − 4x + k is 3 then write the value of k.

By using the relationship between the zeros of the quadratic ploynomial.
We have
Product of zeroes =
$⇒3=\frac{k}{1}\phantom{\rule{0ex}{0ex}}⇒k=3$

Question 13:

If (x + a) is a factor of (2x2 + 2ax + 5x + 10), find the value of a.   [CBSE 2010]

Given: (x + a) is a factor of 2x2 + 2ax + 5x + 10
We have
$x+a=0\phantom{\rule{0ex}{0ex}}⇒x=-a$
Since, (x + a) is a factor of 2x2 + 2ax + 5x + 10
Hence, It will satisfy the above polynomial
$\therefore 2{\left(-a\right)}^{2}+2a\left(-a\right)+5\left(-a\right)+10=0\phantom{\rule{0ex}{0ex}}⇒-5a+10=0\phantom{\rule{0ex}{0ex}}⇒a=2$

Question 14:

If (a − b), a and (a + b) are zeros of the polynomial 2x3 − 6x2 + 5x − 7, write the value of a.

By using the relationship between the zeroes of the cubic ploynomial.
We have
Sum of zeroes =
$⇒a-b+a+a+b=\frac{-\left(-6\right)}{2}\phantom{\rule{0ex}{0ex}}⇒3a=3\phantom{\rule{0ex}{0ex}}⇒a=1$

Question 15:

If x3 + x2ax + b is divisible by (x2x), write the values of a and b.

Equating x2x to 0 to find the zeros, we will get

Since,  x3 + x2ax + b is divisible by x2x.
Hence, the zeros of x2x will satisfy x3 + x2ax + b
$\therefore {\left(0\right)}^{3}+{0}^{2}-a\left(0\right)+b=0\phantom{\rule{0ex}{0ex}}⇒b=0\phantom{\rule{0ex}{0ex}}$

and

Question 16:

If α and β are the zeroes of a polynomial 2x2 + 7x + 5, write the value of α + β + αβ.   [CBSE 2010]

By using the relationship between the zeros of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =

Question 17:

State division algorithm for polynomials.

“If f(x) and g(x) are two polynomials such that degree of f(x) is greater than degree of g(x) where g(x) ≠ 0, then there exists unique polynomials q(x) and r(x) such that

f(x) = g(x) × q(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x).

Question 18:

The sum of the zero and the product of zero of a quadratic polynomial are $\frac{-1}{2}$ and −3 respectively, write the polynomial.

We can find the quadratic polynomial if we know the sum of the roots and product of the roots by using the formula
x2 − (Sum of the zeros)x + Product of zeros
$⇒{x}^{2}-\left(-\frac{1}{2}\right)x+\left(-3\right)\phantom{\rule{0ex}{0ex}}⇒{x}^{2}+\frac{1}{2}x-3\phantom{\rule{0ex}{0ex}}$
Hence, the required polynomial is ${x}^{2}+\frac{1}{2}x-3$.

x22x+13=03x232x+1=0

Question 19:

Write the zeros of the quadratic polynomial f(x) = 6x2 − 3

To find the zeros of the quadratic polynomial we will equate f(x) to 0
$\therefore f\left(x\right)=0\phantom{\rule{0ex}{0ex}}⇒6{x}^{2}-3=0\phantom{\rule{0ex}{0ex}}⇒3\left(2{x}^{2}-1\right)=0\phantom{\rule{0ex}{0ex}}⇒2{x}^{2}-1=0$
$⇒2{x}^{2}=1\phantom{\rule{0ex}{0ex}}⇒{x}^{2}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}⇒x=±\frac{1}{\sqrt{2}}$
Hence, the zeros of the quadratic polynomial f(x) = 6x2 − 3 are $\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}$.

Question 20:

Find the zeros of the quadratic polynomial $f\left(x\right)=4\sqrt{3}{x}^{2}+5x-2\sqrt{3}$

To find the zeros of the quadratic polynomial we will equate f(x) to 0
$\therefore f\left(x\right)=0\phantom{\rule{0ex}{0ex}}⇒4\sqrt{3}{x}^{2}+5x-2\sqrt{3}=0\phantom{\rule{0ex}{0ex}}⇒4\sqrt{3}{x}^{2}+8x-3x-2\sqrt{3}=0\phantom{\rule{0ex}{0ex}}⇒4x\left(\sqrt{3}x+2\right)-\sqrt{3}\left(\sqrt{3}x+2\right)=0$

Hence, the zeros of the quadratic polynomial $f\left(x\right)=4\sqrt{3}{x}^{2}+5x-2\sqrt{3}$ are .

Question 21:

If α and β are the zeroes of a polynomial f(x) = x2 − 5x + k, such that αβ = 1, find the value of k.

By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =

Solving αβ = 1 and α + β = 5, we will get
α = 3 and β = 2
Substituting these values in $\alpha \beta =\frac{k}{1}$, we will get
k = 6

Question 22:

If α and β are the zeroes of a polynomial f(x) = 6x2 + x − 2, find the value of $\left(\frac{\alpha }{\beta }+\frac{\beta }{\alpha }\right)$

By using the relationship between the zeroes of the quadratic ploynomial.
We have,
Sum of zeroes = and Product of zeroes =

Question 1:

Which of the following is a polynomial?

(a) ${x}^{2}-5x+4\sqrt{x}+3$
(b) ${x}^{3/2}-x+{x}^{1/2}+1$
(c) $\sqrt{x}+\frac{1}{\sqrt{x}}$
(d) $\sqrt{2}{x}^{2}-3\sqrt{3}x+\sqrt{6}$

(d) is the correct option.
A polynomial in x of degree n is an expression of the form p(x) =ao +a1x+a2x2 +...+an xn, where an $\ne$0.

Question 2:

Which of the following is not a polynomial?

(a) $\sqrt{3}{x}^{2}-2\sqrt{3}x+5$
(b) $9{x}^{2}-4x+\sqrt{2}$
(c) $\frac{3}{2}{x}^{3}+6{x}^{2}-\frac{1}{\sqrt{2}}x-8$
(d) $x+\frac{3}{x}$

It is because in the second term, the degree of x is −1 and an expression with a negative degree is not a polynomial.

Question 3:

The zeros of the polynomial x2 − 2x − 3 are

(a) −3, 1
(b) −3, −1
(c) 3, −1
(d) 3, 1

Question 4:

The zeros of the polynomial ${x}^{2}-\sqrt{2}x-12$ are
(a) $\sqrt{2},-\sqrt{2}$
(b)
(c)
(d)

Question 5:

The zeros of the polynomial $4{x}^{2}+5\sqrt{2}x-3$ are
(a) $-3\sqrt{2},\sqrt{2}$
(b) $-3\sqrt{2},\frac{\sqrt{2}}{2}$
(c) $\frac{-3\sqrt{2}}{2},\frac{\sqrt{2}}{4}$
(d) none of these

Question 6:

The zeros of the polynomial ${x}^{2}+\frac{1}{6}x-2$ are
(a) −3, 4
(b) $\frac{-3}{2},\frac{4}{3}$
(c) $\frac{-4}{3},\frac{3}{2}$
(d) none of these

Question 7:

The zeros of the polynomial $7{x}^{2}-\frac{11}{3}x-\frac{2}{3}$ are
(a) $\frac{2}{3},\frac{-1}{7}$
(b) $\frac{2}{7},\frac{-1}{3}$
(c) $\frac{-2}{3},\frac{1}{7}$
(d) none of these

Question 8:

The sum and product of the zeros of a quadratic polynomial are 3 and −10 respectively. The quadratic polynomial is

(a) x2 − 3x + 10
(b) x2 + 3x −10
(c) x2 − 3x −10
(d) x2 + 3x + 10

Question 9:

A quadratic polynomial whose zeros are 5 and −3, is

(a) x2 + 2x − 15
(b) x2 − 2x + 15
(c) x2 − 2x − 15
(d) none of these

Question 10:

A quadratic polynomial whose zeros are $\frac{3}{5}$ and $\frac{-1}{2}$, is
(a) 10x2 + x + 3
(b) 10x2 + x − 3
(c) 10x2x + 3
(d) 10x2 x – 3

Multiply by 10, we get
$10{x}^{2}-x-3$

Question 11:

The zeros of the quadratic polynomial x2 + 88x + 125 are

(a) both positive
(b) both negative
(c) one positive and one negative
(d) both equal

Question 12:

If α and β are the zero of x2 + 5x + 8, then the value of (α + β) is

(a) 5
(b) −5
(c) 8
(d) −8

Question 13:

If α and β are the zero of 2x2 + 5x − 8, then the value of (αβ) is

(a) $\frac{-5}{2}$
(b) $\frac{5}{2}$
(c) $\frac{-9}{2}$
(d) $\frac{9}{2}$

Question 14:

If one zero of the quadratic polynomial kx2 + 3x + k is 2, then the value of k is

(a) $\frac{5}{6}$
(b) $\frac{-5}{6}$
(c) $\frac{6}{5}$
(d) $\frac{-6}{5}$

Question 15:

If one zero of the quadratic polynomial (k − 1) x2 + kx + 1 is −4, then the value of k is

(a) $\frac{-5}{4}$
(b) $\frac{5}{4}$
(c) $\frac{-4}{3}$
(d) $\frac{4}{3}$

Question 16:

If −2 and 3 are the zeros of the quadratic polynomial x2 + (a + 1) x + b, then

(a) a = −2, b = 6
(b) a = 2, b = −6
(c) a = −2, b = −6
(d) a = 2, b = 6

Question 17:

If one zero of 3x2 + 8x + k be the reciprocal of the other, then k = ?

(a) 3
(b) −3
(c) $\frac{1}{3}$
(d) $\frac{-1}{3}$

Question 18:

If the sum of the zeros of the quadratic polynomial kx2 + 2x + 3k is equal to the product of its zeros, then k = ?

(a) $\frac{1}{3}$
(b) $\frac{-1}{3}$
(c) $\frac{2}{3}$
(d) $\frac{-2}{3}$

Question 19:

If α, β are the zeros of the polynomial x2 + 6x + 2, then $\left(\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}\right)=?$
(a) 3
(b) −3
(c) 12
(d) −12

Question 20:

If α, β, γ are the zeros of the polynomial x3 − 6x2x + 30, then (αβ + βγ + γα) = ?

(a) −1
(b) 1
(c) −5
(d) 30

Question 21:

If α, β, γ are the zeros of the polynomial 2x3x2 − 13x + 6, then αβγ = ?

(a) −3
(b) 3
(c) $\frac{-1}{2}$
(d) $\frac{-13}{2}$

Question 22:

If α, β, γ be the zeros of the polynomial p(x) such that (α + β + γ) = 3, (αβ + βγ + γα) = −
10 and αβγ = −24, then p(x) = ?

(a) x3 + 3x2 − 10x + 24
(b) x3 + 3x2 + 10x −24
(c) x3 − 3x2 −10x + 24
(d) None of these

Question 23:

If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0, then the third zeros is

(a) $\frac{\mathit{-}\mathit{b}}{\mathit{a}}$
(b) $\frac{b}{a}$
(c) $\frac{c}{a}$
(d) $\frac{-d}{a}$

Question 24:

If one of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0, then the product of the other two zeros is

(a) $\frac{-c}{a}$
(b) $\frac{c}{a}$
(c) 0
(d) $\frac{-b}{a}$

Question 25:

If one of the zeros of the cubic polynomial x3 + ax2 + bx + c is −1, then the product of the other two zeros is

(a) ab − 1
(b) b a − 1
(c) 1 − a + b
(d) 1 + a b

Question 26:

If α, β be the zero of the polynomial 2x2 + 5x + k such that α2 + β2 + αβ = $\frac{21}{4}$, then k = ?
(a) 3
(b) −3
(c) −2
(d) 2

Question 27:

On dividing a polynomial p(x) by a non-zero polynomial q(x), let g(x) be the quotient and r(x) be the remainder, than p(x) = q(x)⋅g(x) + r(x), where

(a) r(x) = 0 always
(b) deg r (x) <deg g(x) always
(c) either r(x) = 0 or deg r(x) <deg g(x)
(d) r(x) = g(x)

Question 28:

Which of the following is a true statement?

(a) x2 + 5x − 3 is a linear polynomial.
(b) x2 + 4x − 1 is a binomial.
(c) x + 1 is a monomial.
(d) 5x3 is a monomial.

Question 1:

Zeros of p(x) = x2 − 2x − 3 are

(a) 1, −3
(b) 3, −1
(c) −3, −1
(d) 1, 3

(b) 3,-1
Here, ${\mathrm{p}\left(\mathrm{x}\right)=x}^{2}-2x-3\phantom{\rule{0ex}{0ex}}$

Question 2:

If α, β, γ are the zeros of the polynomial x3 − 6x2x + 30, then the value of (αβ + βγ + γα) is

(a) −1
(b) 1
(c) −5
(d) 30

(a) −1
Here,

Comparing the given polynomial with , we get:

Question 3:

If α, β are the zeroes of kx2 − 2x + 3k such that α + β = αβ, then k = ?

(a) $\frac{1}{3}$
(b) $\frac{-1}{3}$
(c) $\frac{2}{3}$
(d) $\frac{-2}{3}$

(c) $\frac{2}{3}$
Here, $\mathrm{p}\left(x\right)={x}^{2}-2x+3k$
Comparing the given polynomial with $a{x}^{2}+bx+c$, we get:

It is given that
are the roots of the polynomial.

Also, =$\frac{c}{a}$

Question 4:

It is given that the difference between the zeroes of 4x2 − 8kx + 9 is 4 and k > 0. Then, k = ?

(a) $\frac{1}{2}$
(b) $\frac{3}{2}$
(c) $\frac{5}{2}$
(d) $\frac{7}{2}$

(c) $\frac{5}{2}$
Let the zeroes of the polynomial be .
Here,
p
Comparing the given polynomial with $a{x}^{2}+bx+c$, we get:
a = 4, b = −8k and c = 9
Now, sum of the roots$=-\frac{b}{a}$

Question 5:

Find the zeros of the polynomial x2 + 2x − 195.

Here, p

Question 6:

If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is the reciprocal of the other, find the value of a.

Question 7:

Find a quadratic polynomial whose zeros are 2 and −5.

It is given that the two roots of the polynomial are 2 and −5.
Let
Now, sum of the zeroes, $\mathrm{\alpha }+\mathrm{\beta }$ = 2 + (5) = 3
Product of the zeroes, $\mathrm{\alpha \beta }$ = 2$×$5 = 10
∴ Required polynomial = ${x}^{2}-\left(\mathrm{\alpha }+\mathrm{\beta }\right)x+\mathrm{\alpha \beta }$
$={x}^{2}—\left(-3\right)x+\left(-10\right)\phantom{\rule{0ex}{0ex}}={x}^{2}+3x-10$

Question 8:

If the zeroes of the polynomial x3 − 3x2 + x + 1 are (ab), a and (a + b), find the values of a and b.

The given polynomial and its roots are .

Question 9:

Verify that 2 is a zero of the polynomial x3 + 4x2 − 3x − 18.

Let p$\left(\mathrm{x}\right)={x}^{3}+4{x}^{2}-3x-18$

Question 10:

Find the quadratic polynomial, the sum of whose zeroes is −5 and their product is 6.

Given:
Sum of the zeroes = −5
Product of the zeroes = 6
∴ Required polynomial =
$={x}^{2}-\left(-5\right)x+6\phantom{\rule{0ex}{0ex}}={x}^{2}+5x+6$

Question 11:

Find a cubic polynomial whose zeros are 3, 5 and −2.

Question 12:

Using remainder theorem, find the remainder when p(x) = x3 + 3x2 − 5x + 4 is divided by (x − 2).

Question 13:

Show that (x + 2) is a factor of f(x) = x3 + 4x2 + x − 6.

Question 14:

If α, β, γ are the zeroes of the polynomial p(x) = 6x3 + 3x2 − 5x + 1, find the value of $\left(\frac{1}{\mathrm{\alpha }}+\frac{1}{\mathrm{\beta }}+\frac{1}{\mathrm{\gamma }}\right)$

Comparing the polynomial with ${x}^{3}-{x}^{2}\left(\alpha +\beta +\gamma \right)+x\left(\alpha \beta +\beta \gamma +\gamma \alpha \right)-\alpha \beta \gamma$, we get:

Question 15:

If α, β are the zeros of the polynomial f(x) = x2 − 5x + k such that α − β = 1, find the value of k.

Question 16:

Show that the polynomial f(x) = x4 + 4x2 + 6 has no zeroes.

Question 17:

If one zero of the polynomial p(x) = x3 − 6x2 + 11x − 6 is 3, find the other two zeroes.

Question 18:

If two zeroes of the polynomial p(x) = 2x4 − 3x3 − 3x2 + 6x − 2 are $\sqrt{2}$ and $-\sqrt{2}$, find its other two zeroes.

Question 19:

Find the quotient when p(x) = 3x4 + 5x3 − 7x2 + 2x + 2 is divided by (x2 + 3x + 1).

Question 20:

Use remainder theorem to find the value of k, it being given that when x3 + 2x2 + kx + 3 is divided by (x − 3), then the remainder is 21.