Question

If the polynomials ax³ + 3x² − 13 and 2x³ − 5x + a are divided by (x + 2), leave same remainder, find the value of a.

Answer 1

Let p(x) = ax³ + 3x² − 13 and q(x) = 2x³ − 5x + a

Step 1: Use the Remainder Theorem

For divisor x + 2,

x = −2

So we evaluate:

p(−2) = q(−2)

Step 2: Compute P(−2)

p(x) = ax³ + 3x² − 13

p(−2) = a(−2)³ + 3(−2)² − 13

= a(−8) + 3(4) − 13

= −8a + 12 − 13

= −8a − 1

Step 3: Compute q(−2)

q(x) = 2x³ − 5x + a

q(−2) = 2(−2)³ − 5(−2) + a

= 2(−8) + 10 + a

= −16 + 10 + a

= a − 6

Step 4: Set the remainders equal

−8a − 1 = a − 6

Step 5: Solve for a

−8a − 1 = a − 6

−1 + 6 = a + 8a

5 = 9a

a = `5/9`