Without actual division,prove thata4+2a3-2a2+2a-3 is exactly divisible by the polynomial a2+2a-3 - Maths - Polynomials

Question

Without actual division,prove that
a4+2a3-2a2+2a-3 is exactly
divisible by the polynomial a2+2a-3

Ayush Jha · Accepted Answer

Let,

$p (x) = a^{4} + 2 a^{3} - 2 a^{2} + 2 a - 3 q (x) = a^{2} + 2 a - 3 = (a + 3) (a - 1)$

We can check whether p(x) is completely divisible by q(x) as if

$p (- 3) = 0 a n d p (1) = 0$

So let us check,

$p (- 3) = 81 - 54 - 18 - 6 - 3 = 0 p (1) = 1 + 2 - 2 + 2 - 3 = 0$

So q(x) is a factor of p(x).

Hence proved.

Without actual division,prove that a4+2a3-2a2+2a-3 is exactly divisible by the polynomial a2+2a-3.