# Worksheet: Remainder and Factor Theorem with Synthetic Division

In this worksheet, we will practice identifying factors and zeros and finding the remainder of a polynomial function using the remainder and factor theorem with synthetic division.

Q1:

Liam used synthetic division to prove that 4 is a root of the polynomial . Using his result, factor into three linear factors.

• A
• B
• C
• D
• E

Q2:

Consider the function .

Using synthetic division, find the value of . State which, if any, of and is a factor of .

• AOnly is a factor.
• BBoth and are factors.
• COnly is a factor.
• DNeither nor is a factor.

Q3:

Consider the function .

Given that two of the three numbers , and 3 are roots of , use synthetic division to fully factor .

• A
• B
• C
• D
• E

Q4:

One of the zeros of the function belongs to the set . Using synthetic division, find all the zeros of .

• A−4, 2, 6
• B−4, 3, −5
• C−4, 3, 5
• D4, 3, −5
• E4, 2, −6

Q5:

The function has 2 real zeros and 2 imaginary zeros.

Using synthetic substitution, determine which of the values 1, 2, 3, and 4 is a root of and write in the form .

• A
• B
• C
• D
• E

Using synthetic substitution, determine which of the values , , , and is a root of and write in the form .

• A
• B
• C
• D
• E

State all the zeros of .

• A, 1, ,
• B2, , ,
• C4, , ,
• D4, 1, ,
• E2, , ,

Q6:

A polynomial is divided by . Given that is not a factor of , what is the remainder equal to?

• A
• B
• C0
• D

Q7:

Given that is a factor of , what is the remainder when is divided by ?

Q8:

Consider the function .

Use synthetic division to find the quotient and the remainder satisfying .

• A,
• B,
• C,
• D,
• E,

Find .

Q9:

Use synthetic substitution to find the value of given .

Q10:

Consider the function .

What does the remainder theorem tell us about ?

• A is the remainder when we divide by .
• B is the remainder when we divide by .
• C is the remainder when we divide by .
• D is the remainder when we divide by .

Hence, use synthetic division to find .

Q11:

Find the value of given that is divisible by .