Factor Theorem

The Connection to Zeroes

Factor Theorem: $(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$ .

In other words: Zeroes give us factors!

Using Factor Theorem

Example 1: Is $(x + 2)$ a factor of $x^3 + 3x^2 + 5x + 6$ ?

Step 1: Find the zero of $(x + 2)$ $x + 2 = 0 \Rightarrow x = -2$

Step 2: Calculate $p(-2)$ $p(-2) = (-2)^3 + 3(-2)^2 + 5(-2) + 6$ $= -8 + 12 - 10 + 6$ $= 0$ ✓

Step 3: Conclude Since $p(-2) = 0$ , yes $(x + 2)$ IS a factor! ✓

Example 2: Find $k$ if $(x - 1)$ is factor of $x^2 + x + k$

Step 1: If $(x - 1)$ is factor, then $p(1) = 0$

Step 2: Calculate $p(1)$ $p(1) = (1)^2 + (1) + k = 0$ $1 + 1 + k = 0$ $k = -2$

Answer: $k = -2$ ✓

Factorizing Using Factor Theorem

Steps:

List possible factors (from constant term)
Test each using Factor Theorem
When you find one that works, divide to get quotient
Factor the quotient if possible

Example: Factorize $x^3 - 2x^2 - x + 2$

Step 1: Possible zeroes: ±1, ±2

Step 2: Test $x = 1$ : $p(1) = 1 - 2 - 1 + 2 = 0$ ✓ Found one!

Step 3: $(x - 1)$ is a factor

Step 4: Divide to get quotient: $x^2 - x - 2$

Step 5: Factor quotient: $(x - 2)(x + 1)$

Final Answer: $(x - 1)(x - 2)(x + 1)$

Class 9 Maths (NCERT)

Module 1: Chapter 1: Number Systems

Module 2: Chapter 2: Polynomials

Module 3: Chapter 3: Coordinate Geometry

Module 4: Chapter 4: Linear Equations in Two Variables

Module 5: Chapter 5: Introduction to Euclid's Geometry

Module 6: Chapter 6: Lines and Angles

Module 7: Chapter 7: Triangles

Module 8: Chapter 8: Quadrilaterals

Module 9: Chapter 9: Circles

Module 10: Chapter 10: Heron's Formula

Module 11: Chapter 11: Surface Areas and Volumes

Module 12: Chapter 12: Statistics

Module 13: Chapter 13: Constructions

Module 14: Chapter 14: Probability

Chapter 2: Polynomials

The Connection to Zeroes

Using Factor Theorem

Example 1: Is $(x + 2)$ a factor of $x^3 + 3x^2 + 5x + 6$ ?

Example 2: Find $k$ if $(x - 1)$ is factor of $x^2 + x + k$

Factorizing Using Factor Theorem

Steps:

Example: Factorize $x^3 - 2x^2 - x + 2$

Class 9 Maths (NCERT)

Module 1: Chapter 1: Number Systems

Module 2: Chapter 2: Polynomials

Module 3: Chapter 3: Coordinate Geometry

Module 4: Chapter 4: Linear Equations in Two Variables

Module 5: Chapter 5: Introduction to Euclid's Geometry

Module 6: Chapter 6: Lines and Angles

Module 7: Chapter 7: Triangles

Module 8: Chapter 8: Quadrilaterals

Module 9: Chapter 9: Circles

Module 10: Chapter 10: Heron's Formula

Module 11: Chapter 11: Surface Areas and Volumes

Module 12: Chapter 12: Statistics

Module 13: Chapter 13: Constructions

Module 14: Chapter 14: Probability

Chapter 2: Polynomials

Factor Theorem

The Connection to Zeroes

Using Factor Theorem

Example 1: Is (x+2)(x + 2)(x+2) a factor of x3+3x2+5x+6x^3 + 3x^2 + 5x + 6x3+3x2+5x+6?

Example 2: Find kkk if (x−1)(x - 1)(x−1) is factor of x2+x+kx^2 + x + kx2+x+k

Factorizing Using Factor Theorem

Steps:

Example: Factorize x3−2x2−x+2x^3 - 2x^2 - x + 2x3−2x2−x+2

Example 1: Is $(x + 2)$ a factor of $x^3 + 3x^2 + 5x + 6$ ?

Example 2: Find $k$ if $(x - 1)$ is factor of $x^2 + x + k$

Example: Factorize $x^3 - 2x^2 - x + 2$