Factoring Polynomials Factoring is the process of changing a polynomial with TERMS (things that are added or subtracted) into a polynomial with THINGS THAT ARE BEING MULTIPLIED (factors). For example, 24a 2 b 3 – 36a 4 b a 3 b a 2 b 2 → (12a 2 b 2 ) (2b – 3a 2 b 4 + 4a + 8) 6x 2 + 7x – 5 → (3x + 5 ) ( 2x – 1 ) this thing + this thing – that thing → (something) x (something) THIS IS AN IMPORTANT SKILL, BECAUSE THE ALGEBRA RULES FOR ADDING AND SUBTRACTING THINGS ARE VERY DIFFERENT FROM THE RULES FOR MULTIPLYING.

Factoring Polynomials 1)First, always try to factor out a GCF 2)Second, look for one of THREE “special patterns” a.Difference of two Perfect Squares b.Difference of two Perfect Cubes c.Perfect Square Binomial (three terms) 3)If there are FOUR terms, look to “Factor by Grouping” 4)If there are THREE terms, look to do “Triple Play” or “M.A.G.I.C.” or “the X method”. ALWAYS CHECK YOUR ANSWER, by F.O.I.L., or by using the distributive property !! two terms

Special Patterns # 1 The most common special pattern is called “the difference of two perfect squares”. It only works on BINOMIALS ( expressions with two terms ) Its pattern is a 2 – b 2 = ( a + b ) ( a – b ) There are three basic requirements : 1.The first term must be a perfect square 2.The last term must be a perfect square 3.The sign between them must be subtraction If all three requirements are met, use the pattern: ( sq. root of 1 st + sq. root of 2 nd ) x ( sq. root of 1 st – sq. root of 2 nd )

Example: Factor 25x 2 – 16y 2 : 25x 2 – 16y 2 = (5x + 4y) (5x – 4y) Try these: 1) 36a 2 b 2 – 492) 64x 2 – 100x 4 3) 75x 3 – 3xy 4 4) 16x 4 – 81

Special Patterns # 2 The next most common special pattern is called “perfect square trinomials”. It only works on TRINOMIALS ( expressions with 3 terms ) Its pattern is a 2 ± 2ab + b 2 = ( a ± b ) ( a ± b )=[(a ± b) 2 ] There are three basic requirements : 1.The first term must be a perfect square 2.The last term must be a perfect square 3.The middle term must be DOUBLE the PRODUCT of the square roots of those two perfect square terms. The sign of the second term can be positive or negative, but the sign of the last term MUST be positive If all three requirements are met, use the pattern: ( sq. root of 1 st ± sq. root of 2 nd ) 2

Special Patterns # 3 The least common special pattern is called “the sum or difference of two perfect cubes”. It only works on BINOMIALS ( expressions with two terms ) It has two similar looking patterns. Which one to use depends upon whether the terms are being added (+) or subtracted (–) a 3 – b 3 = ( a – b ) ( a 2 + ab + b 2 ) a 3 + b 3 = ( a + b ) ( a 2 – ab + b 2 ) There are three basic requirements : 1.The first term must be a perfect square 2.The last term must be a perfect square 3.The sign between them can be either subtraction or addition If all three requirements are met, use the correct pattern from above

Factoring by Grouping This method only works on polynomials with an EVEN number of terms (usually four terms) Sometimes this method doesn’t work at all !! 1.Put parenthesis around the first two terms and the last two terms. If the sign in front of the 3 rd term is subtraction, change it to adding a negative 2.Factor out a GCF from each of the two parenthesis. If you cannot factor a GCF, try switching the pairs of terms in the two parenthesis, and then try to factor again. 3.Examine the binomials left inside both of the parenthesis. If they are IDENTICAL, you can continue. If not, you are finished, and the polynomial cannot be factored by grouping ! 4.Re-write the four-term polynomial as two binomials that are being MULTIPLIED: ( 1 st GCF ± 2 nd GCF ) times ( one of the two identical binomials)

Factoring by Grouping Must have an EVEN # of terms 1.Put parenthesis around the first two terms and the last two terms. If the sign in front of the 3 rd term is subtraction, change it to adding a negative 2.Factor out a GCF from each of the two parenthesis. 3.Examine the binomials to see if they are IDENTICAL 4.Re-write it as: ( 1 st GCF ± 2 nd GCF ) ( one of the two identical binomials) 3x 2 + 9xy – 4xz – 12yz (3x 2 + 9xy) + (–4xz – 12yz ) 3x 2 + 9xy + – 4xz – 12yz 3x(x + 3y) + –4z(x + 3y) 3x(x + 3y) – 4z(x + 3y) IDENTICAL (3x – 4z) (x + 3y)

Factoring a Trinomial of the form x 2 ± Bx ± C 1)Write two empty parenthesis to fit the two binomials ( ) ( ) 2)Fill in x & x in the two first spots (x )(x ) 3)Use the two-column method to list ALL of the factors of the “C term”. Pay attention to C’s sign, + or –, and use the correct signs on the factor pairs 4)Determine which one of the factor pairs will ADD to make the “B term”. 5)Write that factor pair, with the correct signs, in the second spaces in each parenthesis. (x ± E )(x ± F ) 6)As always, check your answer with F.O.I.L.

Factoring a Trinomial of the form x 2 ± Bx ± C 1)Write two parenthesis 2)Fill in x & x 3)List ALL of the factors of “C”. 4)Decide which one of the factor pairs will ADD to make “B”. 5)Write that factor pair in the second spaces in each parenthesis. 6)Check your answer with F.O.I.L. x 2 + 7x – 30 ( ) (x ) –30 1 –30 2 –15 3 –10 5 – –10 = –7 ( which is not B, although the 7 part is good), so try switching the signs. – = 7 ( which is B ). Also, –3 x 10 = –30, which is the C term. So, the correct factor pair is –3 and 10 ( x – 3 ) ( x + 10 )

Factoring a Trinomial of the form Ax 2 ± Bx ± C, using M.A.G.I.C. 1) Multiply A times C 2)Use the two-column method to list ALL of the factors of A x C. Pay attention to the sign of A x C, and use the correct signs on the factor pairs. 3)Determine which one of the factor pairs will Add to make the “B term”. 4)Re-write the B term into two terms with x, using the factor pair’s coefficients. Now you will have a four-term polynomial. 5)Factor the four-term polynomial by Grouping (check to see if the two parentheses are Identical.) 6)As always, Check your answer with F.O.I.L.

Factoring a Trinomial of the form Ax 2 ± Bx ± C, using M.A.G.I.C. 1) Multiply A times C 2)List ALL of the factors of A x C. 3)Decide which factor pair will Add to make “B”. 4)Re-write the B term as two terms. 5)Factor by Grouping. 6)Check your answer. 2x x – 40 (2) x (–40) = –80 –80 1 –80 2 –40 4 –20 5 –16 8 –10 –5 16 2x 2 + –5x +16x – 40 (2x 2 – 5x) + (16x – 40) x(2x – 5) + 8(2x – 5) (x + 8) (2x – 5)

Factoring a Trinomial of the form Ax 2 ± Bx ± C, using “Triple Play” 1)Write two sets of empty parenthesis. Write inside the parenthesis (Ax ) and( Ax ), and draw a fraction bar with A underneath it (not Ax, just A) The THREE A’s that you write give this method its name. 2)Multiply A times C 3)Use the two-column method to list ALL of the factors of (A x C). Pay attention to the sign of A x C, and use the correct signs on the factor pairs. 4)Determine which one of the factor pairs will add to make “B”. 5)Write that factor pair in the second places in each parenthesis. 6) Factor any GCF out of one or both parenthesis. Divide the GCF(s) by the bottom number to get rid of the fraction 7)As always, check your answer with F.O.I.L.

Factoring a Trinomial of the form Ax 2 ± Bx ± C, using “Triple Play” 1)Write two sets of empty parenthesis. Write inside the parenthesis (Ax ) and ( Ax ), and draw a fraction bar with A (not Ax, just A) underneath it. 2)Multiply A times C 3)List ALL of the factors of (A x C). 4)Decide which factor pairs will add to make “B”. 5)Write that factor pair in the second places in each parenthesis. 6)Factor any GCF out of one or both parenthesis. 7)Divide the GCF(s) by the bottom number to get rid of the fraction 8)Check your answer with F.O.I.L. 2x x – 40 ( 2x ) ( 2x ) 2 (2) x (–40) = –80 –80 1 –80 2 –40 4 –20 5 –16 8 –10 –5 16 (2x – 5) (x + 8) ( 2x – 5 ) ( 2x + 16 ) 2 ( 2x – 5 ) 2( 2x + 16 ) 2 ( 2x – 5 ) 2( 2x + 16 ) 2 Add to make −11, not +11 Add to make +11 (2x – 5) (x + 8)

Factoring a Trinomial of the form Ax 2 ± Bx ± C, using “Triple Play” 1)Write two sets of empty parenthesis. Write inside the parenthesis (Ax ) and ( Ax ), and draw a fraction bar with A (not Ax, just A) underneath it. 2)Multiply A times C 3)List ALL of the factors of (A x C). 4)Decide which factor pairs will add to make “B”. 5)Write that factor pair in the second places in each parenthesis. 6)Factor any GCF out of one or both parenthesis. 7)Divide the GCF(s) by the bottom number to get rid of the fraction 8)Check your answer with F.O.I.L. 6x 2 – 19x + 8 ( 6x ) ( 6x ) 6 (6) x (8) = –3 – 16 ( 6x – 3 ) ( 6x – 16 ) 6 Add to make +19, not −19 Add to make – 19 ( 6x – 3 ) ( 6x – 16 ) 6 3 ( 2x – 1 ) 2( 3x – 8 ) 3 2 ( 2x – 1 ) ( 3x – 8 )