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Polynomials

Graphical Interpretation of Number of Zeros of a Polynomial

We have learnt to find the zeroes or roots of a polynomial. Also, we know that zeroes or roots of a polynomial are those values of its variables for which the polynomial results to zero.      

Now, let us learn about the relation between zeroes and the coefficients of the polynomial.

Relationship between the zero and the coefficient of a linear polynomial:

Let p(x) = ax + b be a linear polynomial such that a ≠ 0. 

Zero of this polynomial can be obtained by equating it with 0.

i.e,  p(x) = 0

ax + b = 0

ax = –b 

Using this relation, we can find the zero of a linear polynomial.

 

Relationship between the zeroes and the coefficients of a quadratic polynomial:

Consider a quadratic polynomial p(x) = 3x2– 5x – 12.

Can we find out the sum and the product of the zeroes of this polynomial?

Yes, we can find the sum and the product of zeroes but firstly we have to find out the zeroes of the polynomial.

Here, the zeroes of polynomial p(x) are 3 and.

Now, the sum of zeroes = 3 +

And the product of zeroes = 3 × = – 4

Can we find out the sum and the product of zeroes by any other method?

Yes, there is also a method in which there is no need to find out the zeroes. In that method we use the coefficients of the polynomial to find the sum and the product of zeroes.

Firstly let us see the relation between the sum and product of zeroes and the coefficients of the polynomial.

Let us first consider a quadratic polynomial p(x) = ax2 + bx + c, where a, b and c are constants.

If αand β are the zeroes of p(x) = ax2 + bx + c, then,

Sum of zeroes = and

Product of zeroes = αβ =

Now, let us find the sum and product of zeroes of the polynomial given in the beginning, using these relations.

The polynomial is p(x) = 3x2 – 5x – 12.

On comparing this equation with ax2 + bx + c, we have

a = 3, b = –5 and c = –12

∴ Sum of zeroes = –=

And the product of zeroes =

Using these relations we obtained the same values as we found after calculating the zeroes.

Now, let us know the relations between the sum and the product of zeroes and the coefficients of a cubic polynomial.

The general form of a cubic polynomial is p(x) = ax3 + bx2 + cx + d where a, b, c and d are …

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