Dividing Polynomials & the Factor Theorem — Drills
Dividing Polynomials & the Factor Theorem has 5 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills 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.
Remainder
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the remainder when f(x)=1x3 + 0x + 2 is divided by x-5.
Worked answer
127
f(5)=1(125) + 0(5) + 2
=127
Problem 2
Find the remainder when f(x)=2x3 - 2x - 7 is divided by x-4.
Worked answer
113
f(4)=2(64) - 2(4) - 7
=113
Roots
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
What number goes in the box for synthetic division by x-9?
Worked answer
9
x-9=0 gives x=9
Problem 2
What number goes in the box for synthetic division by x-2?
Worked answer
2
x-2=0 gives x=2
Dividing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Divide x2+8x+12 by x+2.
Worked answer
x+6
x2+8x+12=(x+2)(x+6)
So the quotient is x+6
Problem 2
Divide x2+9x+20 by x+4.
Worked answer
x+5
x2+9x+20=(x+4)(x+5)
So the quotient is x+5
Checking a solution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Is x-3 a factor of 3x3 + 4x - 93? Answer yes or no.
Worked answer
yes
f(3)=0
Zero, so it is a factor
Problem 2
Is x-4 a factor of 2x3 - 1x - 120? Answer yes or no.
Worked answer
no
f(4)=4
Not zero, so it is not a factor
Finding the constant
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For what value of k is x-3 a factor of x3+kx-9?
Worked answer
-6
f(3)=27+3k-9=0
3k=-18
k=-6
Problem 2
For what value of k is x-1 a factor of x3+kx-6?
Worked answer
5
f(1)=1+1k-6=0
1k=5
k=5