Polynomial Word Problems

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.

Solved Polynomial Word Problems

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

        

Solved Polynomial Word Problems

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

Solved Polynomial Word Problems

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

Solved Polynomial Word Problems

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

Solved Polynomial Word Problems

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

Solved Polynomial Word Problems

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.