Question

If the polynomials p(x)=x^3+ax^2+5x+2 and q(x)=2x^3−ax+6 leave the same remainder when divided by (x+2), find the value of a.

Answer 1

$\large\underline{\sf{Solution-}}$

Given polynomials are

$\rm :\longmapsto\:p(x) = {x}^{3} + {ax}^{2} + 5x + 2$

and

$\rm :\longmapsto\:q(x) = {2x}^{3} - ax + 6$

Given that,

↝ p(x) and q(x) leaves the same remainder when divided by x + 2.

We know,

Remainder Theorem states that if a polynomial f (x) is divided by linear polynomial x - a, it leaves the remainder f(a).

So, using this concept of Remainder Theorem

$\rm :\longmapsto\:p( - 2) = q( - 2)$

$\rm :\longmapsto\:{( - 2)}^{3} + {a( - 2)}^{2} + 5( - 2) + 2 = 2 {( - 2)}^{3} - 2a + 6$

$\rm :\longmapsto\: - 8 + 4a - 10 + 2 = - 16 + 2a + 6$

$\rm :\longmapsto\: 4a - 16 = 2a - 10$

$\rm :\longmapsto\: 4a - 2a = 16 - 10$

$\rm :\longmapsto\: 2a = 6$

$\rm \implies\:\boxed{ \tt{ \: a \: = \: 3 \: }}$

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Additional Information :-

Factor Theorem states that if x - a is a factor of polynomial f(x), then f(a) = 0

More Identities to know :-

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)