Question

If the polynomials ax³+3x²-13 and 2x³5x+a when divided by (x-2) leave the same remainder, find the value of a.

Answer 1

Given : If the polynomials ax³ + 3x² - 13 and 2x³ - 5x + a when divided by (x - 2) leave the same remainder.

 

Let p (x) = ax³ + 3x² - 13 and q (x) = 2x³ - 5x + a  be the given polynomials. The remainders when p(x) and q(x) are divided by (x - 2) are p (2) and q (2) .

By the given condition, we have :

p(2) = q(2)

⇒ a (2)³ + 3(2)² – 13 = 2 (2)³ – 5(2) + a

⇒ a × 8 + 3 × 4 – 13 = 2 × 8  – 5(2) + a

⇒ 8a + 12 – 13 = 16 – 10 + a

⇒ 8a - 1 = 6 + a

⇒ 8a - a = 6 + 1

⇒ 7a = 7

⇒ 7a = 7

⇒ a = 7/7

⇒ a = 1

Hence, the value of 'a' is 1.

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Answer 2

p(x) = ax³+3x²-13

p'(x) = 2x³-5x+a

g(x) = x-2

Division of p(x) and g(x):

g(x) = 0

x-2 = 0

x = 2

p(2) = a(2)³+3(2)²-13

p(2) = 8a + 12 - 13

p(2) = 8a - 1

Division of p'(x) and g(x):

g(x) = 0

x-2 = 0

x = 2

p'(2) = 2(2)³-5(2)+a

p'(2) = 16 - 10 + a

p'(2) = a + 6

p(2) = p'(2):

8a - 1 = a + 6

=> 7a = 7

=> a = 1