Factoring Polynomials

πŸ§‘Zayn Alazawi πŸ“…July 20, 2020 πŸ“ŠGrade 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 \(a≠1\):

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: \((2x+1)(x+2)\)

III. Introducing the Quadratic Equation:

Quadratic Equation: \(ax^2+bx+c=0\)

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

Use of the Quadratic Formula for Factoring:

  • The quadratic equation helps find the roots (solutions) of a quadratic expression.
  • The expression \(b^2 βˆ’4ac\) is called the discriminant and determines the nature of the roots.

Example: \(x^2βˆ’4x+4=0\)

  • Identify \(a=1\), \(b=βˆ’4\), 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.

✍️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: \((x+3)(x+4)\)

  2. Factor \(3x ^2 βˆ’10x+7\)

    3. Factoring when \(aβ‰ 1\): \(3x ^2 βˆ’10x+7\)

    • Multiply \(a=3\) and \(c=7\): \(ac=21\).
    • Find factors of 21 that add up to -10: \(βˆ’7Γ—βˆ’3=21, βˆ’7+(βˆ’3)=βˆ’10\).
    • Rewrite \(βˆ’10x\) as \(βˆ’7xβˆ’3x\): \(3x^2βˆ’7xβˆ’3x+7\)
    • Group and factor: \((3x^2βˆ’7x)+(βˆ’3x+7)\)
    • Factor: \(x(3xβˆ’7)βˆ’1(3xβˆ’7)\)
    • Factor by grouping: \((3xβˆ’7)(xβˆ’1)\)

  3. Factor \(2x ^2 +3xβˆ’5\)

    4. Factoring when \(aβ‰ 1\): \(2x ^2 +3xβˆ’5\)

    • Multiply \(a=2\) and \(c=βˆ’5\): \(ac=βˆ’10\).
    • Find factors of -10 that add up to 3: \(5Γ—(βˆ’2)=βˆ’10, 5+(βˆ’2)=3\).
    • Rewrite \(3x\) as \(5xβˆ’2x\): \(2x ^2 +5xβˆ’2xβˆ’5\)
    • Group and factor: \((2x^2+5x)+(βˆ’2xβˆ’5)\)
    • Factor: \(x(2x+5)βˆ’1(2x+5)\)
    • Factor by grouping: \((2x+5)(xβˆ’1)\)