Ways to factor…There a few different ways to approaching factoring anexpression. However, the first thing you should alwayslook for is the Greatest Common Factor (GCF). How to find the GCF video Factoring using GCF video
Why GCF first?You look for the GCF first because it will help you factorquadratics using the second method by making thenumbers smaller.The second method of factor involves undoing thedistributive property….I call it unFOILing There are several videos posted on this method, buthere’s an example…..
Factor: x2 + 6x + 8Look at the last number. x2 + 6x + 8If the sign in a positive, the signs in the parenthesis Here the 8 is positive.will be the same.Look at the sign on the middle number.We know the signs will be the same because 8 ispositive. We look a the middle number and its also (x + )(x + )positive. So both signs in the parenthesis will bepositive.Find factors of the last number that when youmulitply them you get that last number, but whenyou combine them you get the middle number. (x + 4)(x + 2)So were looking for factors of 8 that we multiplythem we get an 8, but when we add them we get a6.....4 and 2.Check it with FOIL. (x + 4)(x + 2) = x2 + 4x +2x +8You never get a factoring problem wrong! You canalways check it by multiplying. It works!
Special Case: The Difference of two Perfect SquaresThe difference of two perfect squares is very easy tofactor, but everyone always forgets about them.!Theyrein the form (ax)2 - c where a and c are perfect squares.Theres no visible b-value...so b = 0. You factorthem by taking the square root of a and the square rootof c and placing them in parenthesis that have oppositesigns.Whenever you have a binomial that is subtraction, always check tosee it’s this special case. It usually does NOT have a GCF.Heres an example….
Example Factor : 4x2 – 9Set up parenthesis with opposite signs ( + )( - )Find the square root of a and place thenanswer in the front sections of the ( 2x + )( 2x - )parenthesissqrt(4x2) = 2xFind the square root of c and placethem at the end of the parenthesis. ( 2x + 3 )( 2x - 3 )sqrt(9) = 3 Difference of Two Perfect Squares Video
Practice: Common Factors Practice: Practice:Difference of Two Factor Squares a=1 Factoring with Algebra Tiles
What if the leading coefficient isn’t a 1? Factor: 3x2 + 11x - 4Set up two pairs of parenthesis ( )( )Look over the equation ( + )( - )Look at the a-valueUnfortunately, the a-value is not a one, so we need Factors of A Factors of Cto list factors in a chart. 1, 3 2,2 and 1,4Were looking for the pair of factors that when I 1*2 - 3*2 = -4 NOfind the difference of the products 1*3 - 1*4= -1 NOwill yield the b-value. 1*1 - 3*4= -12 YES!Enter in values (x - 4)(3x + 1)Check with FOIL (x - 4)(3x + 1)= 3x^2 -12x + x - 4Its possible that you have the right numbers but in = 3x^2 -11x -4the wrong spots, so you have to check.
Factoring when a≠ 1Terms in a quadratic expression may have some common factors before you break them down into linear factors. Remember, the greatest common factor, GCF, is the greatest number that is a factor of all terms in the expression. When a ≠ 1, we should always check to see if the quadratic expression has a greatest common factor.
Factor 2x -22x +36 2 Step 1: a ≠ 1, so we should check to see if the quadratic expression has a greatest common factor. It has a GCF of 2. 2x2 -22x +36 = 2(x2 -11x +18) Step 2:Once we factor out the GCF, the quadratic expression now has a value of a =1 and we can use the process we just went through in the previous examples. x2 -11x +18 = (x -2)(x-9) Therefore, 2x2 -22x +36 is = 2 (x -2)(x-9).
A≠ 1 and NO GCF 2x + 13x – 7 2 Step 1: a ≠ 1, so we should check to see if the quadratic expression has a greatest common factor. It does not have a GCF! This type of trinomial is much more difficult to factor than the previous. Instead of factoring the c value alone, one has to also factor the a value. Our factors of a become coefficients of our x-terms and the factors of c will go right where they did in the previous examples.
2x + 13x – 7 2Step 1: Find the product ac. ac= -14Step 2: Find two factors of ac that add to give b. S 1 and -14 = -13 1 -1 and 14 = 13 This is our winner! - 2 and -7 = -5 2 -2 and 7 = 5Step 3: Split the middle term into two terms, using the numbers found in step above. 2x2 -1x + 14x – 7
Step 4: Factor out the common binomial using the box method. 2x2 -1x + 14x – 7Quadratic Factor 1 Term 2x 2 -1x Factor 2 Constant Term 14x -7 Find the GCF for each column and row!
Numbers in RED represent the GCF of each row and column 2x -1 x 2x2 -1x 7 14x -7 The factors are (x + 7)(2x - 1).
Prime FactorsRemember: This won’t work for all quadratic trinomials, because not all quadratic trinomials can be factored into products of binomials with integer coefficients. We call these prime! (Prime Numbers are 3, 5, 7, 11, 13, etc.)Expressions such as x2 + 2x - 7, cannot be factored at all, and is therefore known as a prime polynomial.
Practicing Factoring when a ≠1.Please watch the demonstration below on factoring when a ≠ 1. There will be interactive examples provided to help when a ≠ 1. MORE FACTORING Upon completion of the video anddemonstration, please complete Mastery Assignment Part 2.
Practice: All OtherGizmo: Factoring Cases ax2 + bx + c More Practice: Application Instruction Problems