# Factoring Polynomials

π§Zayn Alazawi πJuly 20, 2020 πGrade 11 Math

I. What is Factoring?

Definition: Factoring is the process of expressing a polynomial as the product of its factors. In other words, it involves breaking down a polynomial into simpler expressions that multiply together to give the original polynomial.

Purpose: Factoring is a crucial skill in algebra because it helps us understand the behavior of functions, solve equations, and manipulate expressions more efficiently.

II. How to Factor Polynomials:

1. Factoring When $$a=1$$:

Quadratic Function: $$f(x)=x^2+bx+c$$

Factoring Process:

1. Identify $$b$$ and $$c$$.
2. Find two numbers whose product is $$c$$ and whose sum is $$b$$.
3. Write $$bx$$ using the two numbers found in the previous step.
4. Factor the expression as the product of two binomials.

Example: $$x^2+5x+6$$

• Identify $$b=5$$ and $$c=6$$.
• Find factors of 6 that add up to 5: $$2Γ3=6, 2 + 3 = 5$$
• Write the factors: $$x^2+2x+3x+6$$
• Factor: $$(x+2)(x+3)$$

2. Factoring When $$aβ 1$$:

Quadratic Function: $$f(x)=ax^2+bx+c$$

Factoring Process:

1. Multiply $$a$$ and $$c$$ to get $$ac$$.
2. Find two numbers whose product is $$ac$$ and whose sum is $$b$$.
3. Split $$bx$$ using the two numbers found in the previous step.
4. Group the terms and factor out the greatest common factor (GCF) from each group.
5. Factor by grouping.

Example: $$2x^2+5x+2$$

• Multiply $$a=2$$ and $$c=2$$: $$ac=4$$.
• Find factors of 4 that add up to 5: $$1Γ4=4, 1 + 4 = 5$$.
• Rewrite $$5x$$ as $$1x+4x$$: $$2x^2+1x+4x+2$$
• Group and factor: $$(2x^2+1x)+(4x+2)$$
• Factor: $$x(2x+1)+2(2x+1)$$
• Factor by grouping: $$(2x+1)(x+2)$$

Quadratic Equation: $$ax^2+bx+c=0$$

Quadratic Formula: $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$ββ

Use of the Quadratic Formula for Factoring:

• The quadratic equation helps find the roots (solutions) of a quadratic expression.
• The expression $$b^2 β4ac$$ is called the discriminant and determines the nature of the roots.

Example: $$x^2β4x+4=0$$

• Identify $$a=1$$, $$b=β4$$, and $$c=4$$.
• Apply the quadratic formula to find the roots.

In summary, factoring is a versatile tool in algebra, allowing us to break down complex expressions and understand the structure of polynomials. Whether οΏ½a is 1 or not, factoring can be achieved through careful identification of terms and application of appropriate techniques. The quadratic equation provides a powerful method for finding the roots of quadratic expressions, aiding in the factoring process.

## βοΈPractice

1. Factor $$x ^2 +7x+12$$

2. Factoring when $$a=1$$: $$x^2+7x+12$$

• Identify $$b=7$$ and $$c=12$$.
• Find factors of 12 that add up to 7: $$3Γ4=12,3+4=7$$.
• Write the factors: $$x^2+3x+4x+12$$
• Factor: $$(x+3)(x+4)$$
3. Factor $$3x ^2 β10x+7$$

4. Factoring when $$aβ 1$$: $$3x ^2 β10x+7$$

• Multiply $$a=3$$ and $$c=7$$: $$ac=21$$.
• Find factors of 21 that add up to -10: $$β7Γβ3=21, β7+(β3)=β10$$.
• Rewrite $$β10x$$ as $$β7xβ3x$$: $$3x^2β7xβ3x+7$$
• Group and factor: $$(3x^2β7x)+(β3x+7)$$
• Factor: $$x(3xβ7)β1(3xβ7)$$
• Factor by grouping: $$(3xβ7)(xβ1)$$
5. Factor $$2x ^2 +3xβ5$$

6. Factoring when $$aβ 1$$: $$2x ^2 +3xβ5$$

• Multiply $$a=2$$ and $$c=β5$$: $$ac=β10$$.
• Find factors of -10 that add up to 3: $$5Γ(β2)=β10, 5+(β2)=3$$.
• Rewrite $$3x$$ as $$5xβ2x$$$$2x ^2 +5xβ2xβ5$$
• Group and factor: $$(2x^2+5x)+(β2xβ5)$$
• Factor: $$x(2x+5)β1(2x+5)$$
• Factor by grouping: $$(2x+5)(xβ1)$$