If the polynomial f(x) = ax3 + bx - c is divisible by the polynomial g(x) = x2+bx+c, then ab - Maths - Polynomials

Question

If the polynomial f(x) = ax3 + bx - c is divisible by
the polynomial g(x) = x2+bx+c, then ab =
a) 1 b) 1/c c) -1 d) -1/c
also, in the above question c =
a) b b) 2b c)2b2 d) -2b
a) b b) 2b c) 2b2 d) -2b

Harleen Birdi · Accepted Answer

Consider, division of f(x) = ax3 + bx â€“ c by g(x) = x2 + bx + c,

Since f(x) is divisible by g(x), remainder must be zero i e , $(a b^{2} - a c + b) x + c (a b - 1) = 0 \Rightarrow (a b^{2} - a c + b) x = 0 and c (a b - 1) = 0 \Rightarrow a b^{2} - a c + b = 0 (∵ x \neq 0) and a b - 1 = 0 (∵ c \neq 0) Now, a b - 1 = 0 \Rightarrow a b = 1$ Consider, $a b^{2} - a c + b = 0 \Rightarrow (a b) b - a c + b = 0 \Rightarrow b - a c + b = 0 (using a b = 1) \Rightarrow 2 b - a c = 0 \Rightarrow c = \frac{2 b}{a} = \frac{2 b^{2}}{a b} \Rightarrow c = \frac{2 b^{2}}{1} (∵ a b = 1) \Rightarrow c = 2 b^{2}$

If the polynomial f(x) = ax3 + bx - c is divisible by the polynomial g(x) = x2+bx+c, then ab = a) 1 b) 1/c c) -1 d) -1/c also, in the above question c = a) b b) 2b c)2b2 d) -2b a) b b) 2b c) 2b2 d) -2b

If the polynomial f(x) = ax³ + bx - c is divisible by the polynomial g(x) = x²+bx+c, then ab =

a) 1 b) 1/c c) -1 d) -1/c

also, in the above question c =

a) b b) 2b c)2b² d) -2b

a) b b) 2b c) 2b2 d) -2b