# Polynomial Word Problems

### Solutions

1Find a and b if the polynomial x5 − ax + b is divisible by x2 − 4.

2Determine the coefficients a and b for the polynomial x3 + ax2 + bx + 5 if it is divisible by x2 + x + 1.

3Find the value of k if the division of 2x2 − kx + 2 by (x − 2) gives a remainder of 4.

4Determine the value of m if 3x2 + mx + 4 has x = 1 as one of its roots.

5Find a fourth degree polynomial that is divisible by x2 − 4 and is annuled by x = 3 and x = 5.

6Calculate the value of a for which the polynomial x3 − ax + 8 has the root x = −2. Also, calculate the other roots of the polynomial.

## 1

Find a and b if the polynomial x5 − ax + b is divisible by x2 − 4.

x2 − 4 = (x +2) · (x − 2)

P(−2) = (−2)5 − a · (−2) + b = 0

−32 +2a +b = 0         2a +b = 32

P(2) = 25 − a · 2 + b = 0

32 − 2a +b = 0          − 2a +b = −32

## 2

Determine the coefficients a and b for the polynomial x3 + ax2 + bx + 5 if it is divisible by x2 + x + 1.

b − a = 0            −a + 6 = 0

a = 6           b = 6

## 3

Find the value of k if the division of 2x2 − kx + 2 by (x − 2) gives a remainder of 4.

P(2) = 2 · 22 − k · 2 +2 = 4

10 − 2k = 4        − 2k = − 6       k = 3

## 4

Determine the value of m if 3x2 + mx + 4 has x = 1 as one of its roots.

P(1) = 3 · 12 + m · 1 + 4 = 0

3 + m + 4 = 0              m = − 7

## 5

Find a fourth degree polynomial that is divisible by x2 − 4 and is annuled by x = 3 and x = 5.

(x − 3) · (x − 5) · (x2 − 4) =

(x2 −8 x + 15) · (x2 − 4) =

= x4 − 4x2 − 8x3 +32x + 15x2 − 60 =

= x4 − 8x3 + 11x2 +32x − 60

## 6

Calculate the value of a for which the polynomial x3 − ax + 8 has the root x = −2. Also, calculate the other roots of the polynomial.

P(−2) = (−2)3 − a · (−2) +8 = 0        −8 + 2a +8 = 0         a= 0

(x + 2) · (x2 − 2x + 4)

x2 − 2x + 4 = 0

It has no more real roots.