Factoring Power Point

By Jason, Blake, Skye and Alicia

CHECK FOR GCF 2 TERMS DIFFERENCE OF SQUARES

3 TERMS

(ax²+bx+c)

TRINOMIAL SQUARE ORDINARY TRINOMIAL (a=1)

COMBINATION TRINOMIALS (a>1)

ex) 16X²-9 =( )( ) =(4X )(4X ) =(4X- )(4X+ ) =(4X-3)(4X+3)

4 TERMS GROUP IN PAIRS

(binomial G.C.F)

DIFFERENCE OF SQUARES – ONE TRI. SQUARE

Definitions & Important Facts •Difference of squares-Is a result when two conjugates are applied to one another. ex)(16x²-9) =(4x-3)(4x+3) • Conjugate-Is the same terms in each set of brackets. However, a different sign is applied to one. ex)(a+b)(a-b) =a²-b² Factoring-Is the process of breaking down numbers into all their factors. ex) 6x²+18x =6x(x+3) •Factor-Quantities multiplied together that produce a product.

•Greatest Common Factor-It is a number that is common in all of the denominators. ex)1\2 + 3\5 =5\10 + 6\10 =11\10 •Monomial-One term ex)2x³ •Binomial- Two terms ex)2x³-1x² •Trinomial-Three terms ex)2x³-1x²+5x •Polynomial-More than three terms ex)2x³-1x²+5x+7 •Double Difference of Squares ex)(a4-16b4) =(a²-4b²)(a²+4b²) =(a-2b)(a+2b)(a²+4b²)

• FOIL-First, Outside, Inside, Last ex)(3x+1)(3x+4) =9x²+12x+3x+4 =9x²+15x+4 • Decomposition-is the process of breaking down a number into smaller units ex)12x²-x-6

ex) 16X²-9 =( )( ) =(4X )(4X ) =(4X- )(4X+ ) FACTORING A DIFFERENCE OF SQUARES:

=(4X-3)(4X+3 G.C.F.

2. If your expression has two terms, check for a difference of squares. If it is a difference of squares, your expression will be a pair of conjugates formed by the square roots. 3. Remove a G.C.F. (if there is one) 4. Make two sets of brackets 5. Take the square root of your first term, and put it as the first term in each of your brackets 6. Next put in the signs, one term will have a negative sign (-), and the other will have a positive sign (+) 7. Now take the square root of the second term of the expression and put it in each bracket as your

ex) 3x²-27 =3(x²-9) =3( )( ) =3(x )(x ) =3(x- )(x+ ) =3(x-3)(x+3)

•The difference between a binomial square and a difference of squares is that the first part of a binomial square is an expression not a single term •Follow the same steps as you would for a regular Expression, ex) (9a-2)²-25 not term difference of =(9a-2-5)(9a-2+5) squares =(9a-7)(9a+3)

•Double difference of squares is basically the same as a difference of squares •In your answer you will have 3 terms

Sum of squares, •There will be a prime, can’t difference of squares, ex) a^4-16b^4 factor and a sum of squares =( )( ) =(a² )(a² )

=(9a-7) 3(3a+1) =3(9a-7)(3a+1)

G.C.F.

=(a²- )(a²+ ) Difference of squares, factor it

=(a²-4b²)(a²-4b²)

=(a-2b)(a+2b) (a²-4b

Factoring Trinomial Squares A trinomial square needs to have: 2. 3.

First and third terms are positive squares Middle term is twice the product of the square roots of the first and third terms.

*The factors of the trinomial square are made from the square roots. When factoring ALWAYS check for a GCF first. If you find that it is a trinomial square then you factor by taking the square root of the first and third term. You put the factors in brackets and then square it. You can check your Example by 1: FOIL. 81-18a+a^2 answer =(9-a)^2

Example 2:

3+12b+12b^2 =3(1+4b+4b^2) =3(1+2b)^2

You may run into some that are prime so you just write prime down as your answer. Example:

y^2+12y+2^4

Factoring Ordinary Trinomials A=1 *Trinomials factor into a pair of binomials

* Check for GCF FIRST. Procedure to factor: x^2-11+24 4. Set up 2 binomials with the factors of x^2 as the first terms. Ex: (x )(x ) 5. Decide the appropriate signs to use in between each term in the parenthesis. Ex: (x- )(x- ) Note: When the product is negative the factors have the opposite sign. When the product is positive they are the same. If they are positive you determine which sign by looking at the sum. If the sum is negative they are both negative, so if the sum is positive they are both positive.

7.

Determine which factors of the product(24) add up to the sum

Factoring Trinomials A>1

You factor these trinomials using decomposition. These are the steps: 4. 5.

3. 9.

12x^2-x-6

Look for a GCF Find 2 integers with a product of (a)(c) and a sum of b. (a is the first term, c is the last term and b is the middle term) Ex: (12)(-6)=-72 b= -1 Factors= 8, -9 Determine the possible first terms. Look at the first term to get these. Ex: (4x )(3x ) To find the rest of the numbers you take the factors that you found and see which of the first terms divide evenly into them. That determines which numbers to use and where they go.

Facto rin g b y Gro uping 4 Term Expressio ns  First, before you do anything, you have to find the Greatest Common Factor (GCF)

Procedure to factor 4-Term Expressions: •

Group into 2 pairs of terms separated by a + sign

•

Remove the GCF from each pair

•

Look for a binomial common factor to remove.

Examples xy + 3x + 2y + 6 = ( xy +3x ) + ( 2y + 6 ) =x(y+3)+2(y+3) =(y+3)(x+2)

1) You group the xy+3x and 2y+6 together 2) You have to make sure to always have a positive sign separating the two expressions 3) Then you take the GCF out of the terms

* Sometimes when a GCF or common binomial factor is not there (when you’re grouping). When this happens you have to rearrange your terms.

4) The terms in the brackets always have to be the same ( positive or negative signs to) 5) Then you finally group everything together the (y+3) goes together and you put the x and the 2 together making it (x+2).

More Examples 1) cx + dy + cy + dx = (cx + dy) + (cy + dx) * There is no GCF to take out… so we rearrange * = (cx + cy) + (dy + dx) = c (x+y) + d (x+y) = (c+d) (x+y)

1) Sometimes when a GCF or common binomial factor is not there (when you’re grouping). When this happens you have to rearrange your terms. *2) Be careful when the third term is negative, you still have to maintain the + sign between the 2 pairs.

2) 6yz – 27z- 16y + 72 = (6yz-27z) + ( -16y+72) = 2y(3z-8) + 9(-3z+8) = * Switch the signs so that the 3 is positive and the 8 is negative. = 2y(3z-8) -9(3z-8) = (2y-9) (3z-8)