# Factoring a polynomial means expressing it as a product of other polynomials.

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Factoring a polynomial means expressing it as a product of other polynomials.

Factoring polynomials with a common monomial factor (using GCF). **Always look for a GCF before using any other factoring method. Factoring Method #1

Steps: 1. Find the greatest common factor (GCF). 2. Divide the polynomial by the GCF. The quotient is the other factor. 3. Express the polynomial as the product of the quotient and the GCF.

Step 1: Step 2: Divide by GCF

The answer should look like this:

Factor these on your own looking for a GCF.

Factoring polynomials that are a difference of squares. Factoring Method #2

A “Difference of Squares” is a binomial ( *2 terms only*) and it factors like this:

To factor, express each term as a square of a monomial then apply the rule...

Here is another example:

End of Day 1

Sum and Difference of Cubes:

Rewrite as cubes Write each monomial as a cube and apply either of the rules. Apply the rule for sum of cubes:

Rewrite as cubes Apply the rule for difference of cubes:

Factoring Method #3 Factoring a trinomial in the form: where a = 1

Next Factoring a trinomial: 2. Find the factors of the c term that add to the b term. For instance, let c = d·e and d+e = b then the factors are (x+d)(x +e ).. 1. Write two sets of parenthesis, (x )(x ). These will be the factors of the trinomial.

x x Factors of +8: 1 & 8 2 & 4 -1 & -8 -2 & -4 Factors of +8 that add to -6 2 + 4 = 6 1 + 8 = 9 -2 - 4 = -6 -1 - 8 = -9 -2 -4

Lets do another example: Find a GCF Factor trinomial Don’t Forget Method #1. Always check for GCF before you do anything else.

When a>1, let’s do something different! Step 1: Multiply a · c Step 2: Find the factors of a·c (-30) that add to the b term = - 30

Factors of 6 · (-5) : 1, -30 1+-30 = -29 -1, 30 -1+30 = 29 2, -15 2+-15 =-13 -2, 15 -2+15 =13 3, -10 3+ -10 =-7 -3, 10 -3+ 10 =7 5, -6 5+ -6 = -2 -5, 6 -5+6 =1 Step 2: Find the factors of a·c that add to the b term Let a·c = d and d = e·f then e+f = b d = -30 e = -2 f = 15

-2, 15 -2+15 =13 Step 3: Rewrite the expression separating the b term using the factors e and f Step 4: Group the first two and last two terms.

Step 4: Group the first Two and last two terms. Step 5: Factor GCF from each group Check!!!! If you cannot find two common factors, Then this method does not work. Step 6: Factor out GCF Common factors

Step 3: Place the factors inside the parenthesis until O + I = bx. FOIL O + I = 30 x - x = 29x This doesn’t work!! Try: I am not a fan of guess and check!

FOIL O + I = -6x + 5x = -x This doesn’t work!! Switch the order of the second terms and try again.

Try another combination: FOIL O+I = 15x - 2x = 13x IT WORKS!! Switch to 3x and 2x

Factoring Technique #3 continued Factoring a perfect square trinomial in the form:

Perfect Square Trinomials can be factored just like other trinomials (guess and check), but if you recognize the perfect squares pattern, follow the formula!

Does the middle term fit the pattern, 2ab? Yes, the factors are (a + b) 2 : b a

Does the middle term fit the pattern, 2ab? Yes, the factors are (a - b) 2 : b a

Factoring Technique #4 Factoring By Grouping for polynomials with 4 or more terms

Factoring By Grouping 1. Group the first set of terms and last set of terms with parentheses. 2. Factor out the GCF from each group so that both sets of parentheses contain the same factors. 3. Factor out the GCF again (the GCF is the factor from step 2).

Step 1: Group Example 1: Step 2: Factor out GCF from each group Step 3: Factor out GCF again

Example 2: