Dividing Polynomials & the Factor Theorem — Practice
Dividing polynomials is the long division you already know, and a remainder of zero names a factor.
What these questions cover
These questions use 7 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rational — Any number writable as ab with integers a and b, b ≠ 0.
- Factor — A number that divides evenly into another.
- Roots — The x values where y = 0.
- Divisor — The polynomial you are dividing by.
- Remainder — What is left when the divisor no longer fits. Always lower degree than the divisor.
- Remainder theorem — Dividing f(x) by x - a leaves remainder f(a).
- Factor theorem — If f(a) = 0 then x - a is a factor, and the other way round.
Problem 1
Find the remainder when f(x)=x2+3x-5 is divided by x-1.
Worked answer
-1
f(1)=1+3-5=-1
Problem 2
Find the remainder when f(x)=x3-2x+4 is divided by x-2.
Worked answer
8
f(2)=8-4+4=8
Problem 3
What value goes in the synthetic division box when dividing by x+5?
Worked answer
-5
x+5=0 gives x=-5
Problem 4
What value goes in the box when dividing by x-7?
Worked answer
7
x-7=0 gives x=7
Problem 5
If f(3)=0, which of these is a factor of f(x)?
- x-3
- x+3
- 3x
- x
Worked answer
x-3
A root at 3 means x-3 divides exactly
Problem 6
Divide x2+7x+12 by x+3.
Worked answer
x+4
x2 ÷ x = x
Multiply back and subtract to get 4x+12
4x ÷ x = 4, remainder 0
Problem 7
Divide x2-x-6 by x-3.
Worked answer
x+2
x2-x-6=(x-3)(x+2)
Problem 8
Divide 2x2+5x-3 by x+3.
Worked answer
2x-1
2x2 ÷ x = 2x
2x(x+3)=2x2+6x, subtract to get -x-3
-x ÷ x = -1, remainder 0
Problem 9
Is x-2 a factor of x3-3x2+4? Answer yes or no.
Worked answer
yes
f(2)=8-12+4=0
A zero remainder means yes
Problem 10
Is x+1 a factor of x3+2x2-5? Answer yes or no.
Worked answer
no
f(-1)=-1+2-5=-4
Not zero, so no
Problem 11
Divide x3-8 by x-2.
Worked answer
x^2+2x+4
Synthetic division with 2: 1, 2, 4, remainder 0
Problem 12
Factor x3-7x+6 completely. Give the three roots, smallest first, separated by commas.
Worked answer
-3,1,2
f(1)=1-7+6=0
Dividing gives x2+x-6=(x+3)(x-2)
Roots -3, 1, 2
Problem 13
One root of x3-4x2+x+6 is -1. Find the other two, smallest first.
Worked answer
2,3
Dividing gives x2-5x+6
(x-2)(x-3)
Problem 14
For what value of k is x-2 a factor of x3+kx-6?
Worked answer
-1
8+2k-6=0
2k=-2 so k=-1
Problem 15
Find the remainder when 2x3-3x2+x-5 is divided by x+1.
Worked answer
-11
2(-1)-3(1)+(-1)-5
=-2-3-1-5=-11
Problem 16
Which list contains every possible rational root of x3-2x2+x-6?
- ±1,±2,±3,±6
- ±1,±2
- ±1,±6
- ±2,±3
Worked answer
±1,±2,±3,±6
The constant is -6
Its divisors are 1, 2, 3, 6, positive and negative