2.7.1 Factor & Remainder Theorem

Factor Theorem

What is the factor theorem?

The factor theorem is used to find the linear factors of polynomial equations
This topic is closely tied to finding the zeros and roots of a polynomial function/equation
- As a rule of thumb a zero refers to the polynomial function and a root refers to a polynomial equation

For any polynomial function P(x)
- (x - k) is a factor of P(x) if P(k) = 0
- P(k) = 0 if (x - k) is a factor of P(x)

How do I use the factor theorem?

Consider the polynomial function P(x) = a_nxⁿ+ a_n-1xⁿ^-1+ … + a₁x + a₀ and (x - k) is a factor

Then, due to the factor theorem P(k) = a_nkⁿ+ a_n-1kⁿ^-1+ … + a₁k + a₀ = 0
$P (x) = (x - k) \times Q (x)$ , where Q(x) is a polynomial that is a factor of P(x)
Hence, $\frac{P (x)}{x - k} = Q (x)$ , where Q(x) is another factor of P(x)

If the linear factor has a coefficient of x then you must first factorise out the coefficient

If the linear factor is $(a x - b) = a (x - \frac{b}{a}) \to P (\frac{b}{a}) = 0$

Exam Tip

A common mistake in exams is using the incorrect sign for either the root or the factor
If you are asked to find integer solutions to a polynomial then you only need to consider factors of the constant term

Worked Example

Determine whether $(x - 2)$ is a factor of the following polynomials:

f (x) = x^{3} - 2 x^{2} - x + 2

g (x) = 2 x^{3} + 3 x^{2} - x + 5

2-7-1-ib-aa-hl-factor-theorem-b-we-solution

It is given that $(2 x - 3)$ is a factor of $h (x) = 2 x^{3} - b x^{2} + 7 x - 6$ .

Find the value of

b

mZEjMdDm_2-7-1-ib-aa-hl-factor-theorem-c-we-solution

Remainder Theorem

What is the remainder theorem?

The remainder theorem is used to find the remainder when we divide a polynomial function by a linear function
When any polynomial P(x) is divided by any linear function (x - k) the value of the remainder R is given by P(k) = R
- Note, when P(k) = 0 then (x - k) is a factor of P(x)

How do I use the remainder theorem?

Consider the polynomial function P(x) = a_nxⁿ+ a_n-1xⁿ^-1+ … + a₁x + a₀ and the linear function (x - k)

Then, due to the remainder theorem P(k) = a_nkⁿ+ a_n-1kⁿ^-1+ … + a₁k + a₀ = R
$P (x) = (x - k) \times Q (x) + R$ , where Q(x) is a polynomial
Hence, $\frac{P (x)}{x - k} = Q (x) + \frac{R}{x - k}$ , where R is the remainder

If the linear factor has a coefficient of x then you must first factorise out the coefficient

If the linear factor is $(a x - b) = a (x - \frac{b}{a}) \to P (\frac{b}{a}) = R$

Worked Example

Let $f (x) = 2 x^{4} - 2 x^{3} - x^{2} - 3 x + 1$ , find the remainder $R$ when $f (x)$ is divided by:

x - 3

2-7-1-ib-aa-hl-remainder-theorem-a-we-solution

x + 2

2-7-1-ib-aa-hl-remainder-theorem-b-we-solution The remainder when $f (x)$ is divided by $(2 x + k)$ is $\frac{893}{8}$ .

Given that

k > 0

, find the value of

k

2-7-1-ib-aa-hl-remainder-theorem-c-we-solution

DP IB Maths: AA HL

Revision Notes

2.7.1 Factor & Remainder Theorem

Factor Theorem

What is the factor theorem?

How do I use the factor theorem?

Exam Tip

Worked Example

Remainder Theorem

What is the remainder theorem?

How do I use the remainder theorem?

Worked Example

Author: Lucy

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