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CIE A-Level Maths Study Notes

2.1.3. Factor Theorem and Remainder Theorem

In this section, the focus is on the Factor Theorem and Remainder Theorem, crucial concepts in algebra for solving polynomial equations. These theorems enable us to analytically find factors of polynomials, evaluate unknown coefficients, and solve polynomial equations with precision.

The Remainder Theorem

The Remainder Theorem posits that when a polynomial f(x)f(x) is divided by a linear term xax - a, the remainder is f(a)f(a). This theorem simplifies the process of evaluating polynomials at specific points, eliminating the need for complex division.

Example: Evaluating a Polynomial

Consider the polynomial f(x)=x3+4x25x14f(x) = x^3 + 4x^2 - 5x - 14. To find the remainder when dividing by x2x - 2, simply evaluate f(2)f(2):

f(2)=23+4(2)25(2)14=8+161014=0.f(2) = 2^3 + 4(2)^2 - 5(2) - 14 = 8 + 16 - 10 - 14 = 0.

Thus, the remainder is 0.

The Factor Theorem

The Factor Theorem extends the Remainder Theorem. It states that xax - a is a factor of a polynomial f(x)f(x) if and only if f(a)=0f(a) = 0. Essentially, aa is a root of the polynomial.

Example: Factorising a Polynomial

Using the polynomial f(x)=x3+4x25x14f(x) = x^3 + 4x^2 - 5x - 14, and since f(2)=0f(2) = 0, we deduce (x2)(x - 2) is a factor. To find the complete factorisation, we divide f(x)f(x) by (x2)(x - 2):

factoring polynomial

Hence, f(x)=(x2)(x2+6x+7)f(x) = (x - 2)(x^2 + 6x + 7).

Solving Polynomial Equations

Example:

Solve the polynomial equation x3+4x25x14=0x^3 + 4x^2 - 5x - 14 = 0, given that x=2x = 2 is a root.

Solution:

1. Given Polynomial: x3+4x25x14=0x^3 + 4x^2 - 5x - 14 = 0

2. Root: x=2x = 2

3. Factor Theorem: Using x=2x = 2 to factorize, we get the quadratic equation x2+6x+7x^2 + 6x + 7.

4. Quadratic Formula: To find the other roots, apply x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=1,b=6,c=7a = 1, b = 6, c = 7.

5. Calculate Roots: Substitute values into the quadratic formula to find the remaining roots.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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