π§Zayn Alazawi π July 20, 2020 πGrade 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:**

- Identify \(b\) and \(c\).
- Find two numbers whose product is \(c\) and whose sum is \(b\).
- Write \(bx\) using the two numbers found in the previous step.
- 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: \(2Γ3=6, 2 + 3 = 5 \)
- 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:**

- Multiply \(a\) and \(c\) to get \(ac\).
- Find two numbers whose product is \(ac\) and whose sum is \(b\).
- Split \(bx\) using the two numbers found in the previous step.
- Group the terms and factor out the greatest common factor (GCF) from each group.
- 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: \(1Γ4=4, 1 + 4 = 5\).
- 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: \((2x+1)(x+2)\)

**III. Introducing the Quadratic Equation:**

**Quadratic Equation:** \(ax^2+bx+c=0\)

**Quadratic Formula:** \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)ββ

**Use of the Quadratic Formula for Factoring:**

- The quadratic equation helps find the roots (solutions) of a quadratic expression.
- The expression \(b^2 β4ac\) is called the discriminant and determines the nature of the roots.

**Example:** \(x^2β4x+4=0\)

- Identify \(a=1\), \(b=β4\), 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.

Factor \(x ^2 +7x+12\)

- Identify \(b=7\) and \(c=12\).
- Find factors of 12 that add up to 7: \(3Γ4=12,3+4=7\).
- Write the factors: \(x^2+3x+4x+12\)
- Factor: \((x+3)(x+4)\)
Factor \(3x ^2 β10x+7\)

- Multiply \(a=3\) and \(c=7\): \(ac=21\).
- Find factors of 21 that add up to -10: \(β7Γβ3=21, β7+(β3)=β10\).
- Rewrite \(β10x\) as \(β7xβ3x\): \(3x^2β7xβ3x+7\)
- Group and factor: \((3x^2β7x)+(β3x+7)\)
- Factor: \(x(3xβ7)β1(3xβ7)\)
- Factor by grouping: \((3xβ7)(xβ1)\)
Factor \(2x ^2 +3xβ5\)

- Multiply \(a=2\) and \(c=β5\): \(ac=β10\).
- Find factors of -10 that add up to 3: \(5Γ(β2)=β10, 5+(β2)=3\).
- Rewrite \(3x\) as \(5xβ2x\): \(2x ^2 +5xβ2xβ5\)
- Group and factor: \((2x^2+5x)+(β2xβ5)\)
- Factor: \(x(2x+5)β1(2x+5)\)
- Factor by grouping: \((2x+5)(xβ1)\)

Factoring when \(a=1\): \(x^2+7x+12\)

Factoring when \(aβ 1\): \(3x ^2 β10x+7\)

Factoring when \(aβ 1\): \(2x ^2 +3xβ5\)