# Polynomial Word Problems

### Solutions

1Find * a* and

*if the polynomial x*

**b**^{5}− ax + b is divisible by x

^{2}− 4.

2Determine the coefficients * a* and

*for the polynomial x*

**b**^{3}+ ax

^{2}+ bx + 5 if it is divisible by x

^{2}+ x + 1.

3Find the value of * k* if the division of 2x

^{2}− kx + 2 by (x − 2) gives a remainder of 4.

4Determine the value of * m* if 3x

^{2}+ mx + 4 has x = 1 as one of its roots.

5Find a fourth degree polynomial that is divisible by x^{2} − 4 and is annuled by x = 3 and x = 5.

6Calculate the value of * a* for which the polynomial x

^{3}− ax + 8 has the root x = −2. Also, calculate the other roots of the polynomial.

## Solved Polynomial Word Problems

## 1

Find * a* and

*if the polynomial x*

**b**^{5}− ax + b is divisible by x

^{2}− 4.

x^{2} − 4 = (x +2) · (x − 2)

P(−2) = (−2)^{5} − a · (−2) + b = 0

−32 +2a +b = 0 2a +b = 32

P(2) = 2^{5} − a · 2 + b = 0

32 − 2a +b = 0 − 2a +b = −32

## Solved Polynomial Word Problems

## 2

Determine the coefficients * a* and

*for the polynomial x*

**b**^{3}+ ax

^{2}+ bx + 5 if it is divisible by x

^{2}+ x + 1.

b − a = 0 −a + 6 = 0

a = 6 b = 6

## Solved Polynomial Word Problems

## 3

Find the value of * k* if the division of 2x

^{2}− kx + 2 by (x − 2) gives a remainder of 4.

P(2) = 2 · 2^{2} − k · 2 +2 = 4

10 − 2k = 4 − 2k = − 6 k = 3

## Solved Polynomial Word Problems

## 4

Determine the value of * m* if 3x

^{2}+ mx + 4 has x = 1 as one of its roots.

P(1) = 3 · 1^{2} + m · 1 + 4 = 0

3 + m + 4 = 0 m = − 7

## Solved Polynomial Word Problems

## 5

Find a fourth degree polynomial that is divisible by x^{2} − 4 and is annuled by x = 3 and x = 5.

(x − 3) · (x − 5) · (x^{2} − 4) =

(x^{2} −8 x + 15) · (x^{2} − 4) =

= x^{4} − 4x^{2} − 8x^{3} +32x + 15x^{2} − 60 =

= x^{4} − 8x^{3} + 11x^{2} +32x − 60

## Solved Polynomial Word Problems

## 6

Calculate the value of * a* for which the polynomial x

^{3}− 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) · (x^{2} − 2x + 4)

x^{2} − 2x + 4 = 0

It has no more real roots.