The Factor and Remainder Theorem form a small, but significant part of the course. It is not at all difficult, but it is worth reminding yourself of these theorem, since, in examinations students often forget them and waste a lot of time using long methods instead. The Factor Theorem is a theorem that allows us to find factors of polynomial functions, to find zeros and ultimately to help us sketch graphs.
A polynomial function f(x) is an algebraic expression that takes the form
\({ f(x)=a }_{ n }{ x }^{ n }+{ a }_{ n-1 }{ x }^{ n-1 }+{ a }_{ n-2 }{ x }^{ n-2 }+\quad ...\quad +{ a }_{ 1 }{ x }^{ 1 }+{ a }_{ 0 }\)
For example, a polynomial function of degree 3 is a cubic function e.g \(f(x) = 7x^3+4x^2-3x+4\)
On this page, you should learn about
- polynomial functions
- zeros, roots and factors
- the factor and remainder theorem
The following videos will help you understand all the concepts from this page
The Remainder Theorem states that for a polynomial f(x),
the remainder when divided by (x-a) is f(a).
Example
Let \(f(x)=2x^3+4x^2+x-5\)
If we divide this polynomial by \((x-2)\)...
\(\frac{2x^3+4x^2+x-5}{x-2}=2x^2+8x+17+\frac{29}{x-2}\)
... we get a quotient \(2x^2+8x+17\) and a remainder 29
When we work out f(2)...
\(f(2)=2(2)^3+4(2)^2+(2)-5=29\)...
... we get 29
The factor theorem states that for a polynomial f(x),
(x-a) is a factor if and only if f(a)=0
Example
Let \(f(x)=x^4+2x^3-7x^2-8x+12\)
When we work out f(1)...
\(f(1)=(1)^4+2(1)^3-7(1)^2-8(1)+12 = 0\)
If we divide this polynomial by \((x-1)\)...
\(\frac{x^4+2x^3-7x^2-8x+12}{x-1}=x^3+3x^2-4x-12\)
... there is no remainder, \((x-1)\) is a factor of the polynomial
In this video, we will look at an application of the factor theorem which is to factorize polynomials:
Factorize completely \(f(x)=2x^3+x^2-7x-6\)
Notes from the video
In the following video, we will look at a typical exam-style question:
The cubic polynomial \(f(x)=ax^3+bx^2-29x+60\) has a factor \((x+4)\) and leaves a remainder of 6 when divided by \((x-2)\).
a. Find the value of a and b.
b. Factorize the polynomial
Notes from the video
Print from here
Here is a quiz that practises the skills from this page
START QUIZ!
Given that f(x) = x3+6x2+14x+15 , find f(1)
f(1) =
f(1) = 13+6(1)2+14(1)+15 = 36
Given that f(x) = x3+6x2+14x+15 , find f(-3)
f(-3) =
f(-3) = (-3)3+6(-3)2+14(-3)+15 = -27 + 54 - 42 + 15 = 0
note that x - 3 would be a factor of this polynomial
Find the remainder when x3 + 2x2 - 5x - 6 is divided by x - 1
Remainder =
Let f(x) = x3 +2x2 -5x - 6
Remainder = f(1) = (1)3 +2(1)2 - 5(1) - 6 = 1 + 2 - 5 - 6 = -8
Find the remainder when x4 - 3x3 - 7x2 + 15x + 18 is divided by x + 2
Remainder =
Remainder = f(-2) = (-2)4 - 3(-2)3 - 7(-2)2 + 15(-2) + 18 = 16 + 24 - 28 - 30 +18 = 0
Hence x + 2 is a factor.
f(x) is a polynomial of degree 3
f(1) = 3
f(2) = -4
f(1.5) = 0
Which of the following statements is definitely true?
If \(f(\frac {3}{2})=0\) then \((x - \frac{3}{2})\) is a factor or (2x - 3)
f(x) is a polynomial of degree 3 and f(-1) = f(1) = f(2) = 0.
Which of the following could be the function f(x)?
f(-1) = 0 implies x + 1 is a factor
f(1) = 0 implies x - 1 is a factor
f(2) = 0 implies x - 2 is a factor
f(x) = (x + 1)(x - 1)(x - 2)
f(x) =(x² - 1)(x - 2)
f(x) is a polynomial of degree 4
f(-1) = f(2) = f(3) = 0
Which of the following could be a graph of f
A | B |
C | D |
The key piece of information is that the polynomial is of degree 4
Hence it has 4 roots with one repeated
The polynomial f(x) = 3x3 + ax2 + x - 10 is divisible by (x - 2). Find the value of a.
a =
If f(x) is divisible by (x - 2) , then (x - 2) is a factor.
f(2) = 3(2)3 + a(2)2 + (2) - 10 = 0
24 + 4a + 2 - 10 = 0
4a = -16
a = -4
When the polynomial f(x) = x4 + 4x3 - 2x2 - 12x + a is divided by (x + 1) it leaves a remainder of 16.
Find a.
a =
f(-1) = (-1)4 + 4(-1)3 - 2(-1)2 - 12(-1) + a = 16
1 - 4 - 2 + 12 + a = 16
a +7 = 16
a = 9
Complete the gaps to make the following statements true
Put rounded brackets around your factor
3x3 - x2 - 20x - 12 = (x² - x - 6)
2x3 + 5x2 - 2x - 5 = (x² - 1)
4x4 - 15x3 - 7x2 + 60x - 36 = x3 - 3x2 - 4x + 12
The cubic polynomial \(3x^3+ax^2+bx-12\) has a factor \((x-2)\) and leaves a remainder of -20 when divided by \((x-1)\).
Find the value of a and b.
Hint
Full Solution
Given that \(ax^3+bx^2+17x-6\) is exactly divisible by \((x-1)(x-2)\), find the value of a and b.
Hint
Full Solution
It is given that \(f(x)=x^3+ax^2+bx+8\)
a. Given that \(x^2-4\) is a factor of \(f(x)\), find the values of a and b.
b. Factorize \(f(x)\) into a product of linear factors.
c. Sketch the graph of \(y=f(x)\) labelling any stationary points and the x and y intercepts.
d. Hence state the range of values of k for which \(f(x)=k\) has exactly one root.
Hint
Full Solution
How much of Factor and Remainder Theorem have you understood?
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