# CHECK IF THE GIVEN IS ZERO OF POLYNOMIAL WORKSHEET

For each of the following cubic equations one root is given. Determine the other roots of each cubic.

Problem 1 :

x3 + 3x2 – 6x – 8 = 0 has a root at x = 2.

Solution

Problem 2 :

x3 + 2x2 – 21x + 18 = 0 has a root at x = 3.

Solution

Problem 3 :

x3 + 4x2 + 7x + 6 = 0 has a root at x = -2.

Solution

Problem 4 :

2x3 + 9x2 + 3x – 4 = 0 has a root at x = -4.

Solution

For each of the following cubic equations use synthetic division to determine if the given value of x is a root of the equation. Where it is, determine the other roots of the equation.

Problem 5 :

Is x = -2 a root of the equation x3 + 9x2 + 26x + 24 = 0 ?

Solution

Problem 6 :

Is x = 4 a root of the equation x3 - 6x2 + 9x + 1 = 0 ?

Solution

Problem 7 :

Is x = -1 a root of the equation x3 + 6x2 + 3x - 5 = 0 ?

Solution

Problem 8 :

Is x = 2 a root of the equation x3 + 2x2 - 20x + 24 = 0 ?

Solution

1)  Other root are x = -1, -4 and 2.

2)  Other roots are x = -6, 1 and 3.

3) Other roots are x = 3, -1 and -2.

4)  Other roots are x = 1/2, -1 and -4.

5)  Other roots are x = -3, -4 and -2.

6)  x = 4 is not a root.

7)  x = -1 is not a root.

8)  Roots are x = 2, -6 and 2.

Use the remainder theorem to find f(k).

Problem 1 :

k = 2; f(x) = x² - 2x + 5

A) -5     B) -3    C) -13     D) 5

Solution

Problem 2 :

k = -3; f(x) = x² + 2x + 2

A) 1     B) -13     C) 5     D) -17

Solution

Problem 3 :

k = -2; f(x) = 3x³ - 7x² - 3x + 3

A) 14     B) -55     C) -43     D) -5

Solution

Problem 4 :

k = 4; f(x) = x³ - 2x² + 5x - 2

A) 54     B) 50     C) -78     D) -76

Solution

Problem 5 :

k = 2; f(x) = 9x4 + 10x³ + 6x² - 6x + 16

A) 360     B) 500     C) 252     D) 36

Solution

Problem 6 :

k = 5; f(x) = x³ - 3x² - 4x - 5

A) 35     B) 25    C) -225     D) -220

Solution

Problem 7 :

Using the remainder theorem find the remainders obtained when x3 + (kx + 8)x + k is divided by x + 1 and x - 2. Hence, find k if the sum of the two remainders is 1.

Solution

1)  Remainder = 5

2)  Remainder = 5

3)  Remainder = -43

4)  Remainder = 50

5)  Remainder = 252

6)  Remainder = 25

7)  k = -2

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