INTRODUCTION TO FACTORING POLYNOMIALS

INTRODUCTION TO FACTORING POLYNOMIALS

MSJC ~ San Jacinto CampusMath Center Workshop SeriesJanice Levasseur

DefinitionsRecall: Factors of a number are the numbers that divide the original number evenly.Writing a number as a product of factors is called a factorization of the number.The prime factorization of a number is the factorization of that number written as a product of prime numbers.Common factors are factors that two or more numbers have in common.The Greatest Common Factor (GCF) is the largest common factor.

Ex: Find the GCF(24, 40).Prime factor each number:24402122623 24 = 2*2*2*3 = 23*322021025 40 = 2*2*2*5 = 23*5 GCF(24,40)= 23 = 8

The Greatest Common Factor of terms of a polynomial is the largest factor that the original terms shareEx: What is the GCF(7x2, 3x) 7x2 = 7 * x * x 3x = 3 * x The terms share a factor of x GCF(7x2, 3x) = x

Ex: Find the GCF(6a5,3a3,2a2)6a5 = 2*3*a*a*a*a*a3a3 = 3*a*a*a2a2 = 2*a*aThe terms share two factors of a GCF(6a5,3a3,2a2)= a2Note: The exponent of the variable in the GCF is the smallest exponent of that variable the terms

DefinitionsTo factor an expression means to write an equivalent expression that is a productTo factor a polynomial means to write the polynomial as a product of other polynomialsA factor that cannot be factored further is said to be a prime factor (prime polynomial)A polynomial is factored completely if it is written as a product of prime polynomials

To factor a polynomial completely, askDo the terms have a common factor (GCF)?Does the polynomial have four terms? Is the polynomial a special one?Is the polynomial a difference of squares?a2 b2Is the polynomial a sum/difference of cubes?a3 + b3 or a3 b3Is the trinomial a perfect-square trinomial?a2 + 2ab + b2 or a2 2ab + b2Is the trinomial a product of two binomials?Factored completely?

Ex: Factor 7x2 + 3xThink of the Distributive Law: a(b+c) = ab + ac reverse it ab + ac = a(b + c) Do the terms share a common factor? What is the GCF(7x2, 3x)?Recall: GCF(7x2, 3x) = x7x2+3x=x( + )Whats left?xxFactor out 7x2 + 3x = x(7x + 3)

Ex: Factor 6a5 3a3 2a2Recall: GCF(6a5,3a3,2a2)= a26a5 3a3 2a2 = a2( - - )a2a2a2316a33a2 6a5 3a3 2a2 = a2(6a3 3a 2)

Your Turn to Try a Few

Ex: Factor x(a + b) 2(a + b)Always ask first if there is common factor the terms share . . . x(a + b) 2(a + b)Each term has factor (a + b) x(a + b) 2(a + b) = (a + b)( )(a + b)(a + b)x2 x(a + b) 2(a + b) = (a + b)(x 2)

Ex: Factor a(x 2) + 2(2 x)As with the previous example, is there a common factor among the terms? Well, kind of . . . x 2 is close to 2 - x . . . Hum . . . Recall: (-1)(x 2) = - x + 2 = 2 x a(x 2) + 2(2 x) = a(x 2) + 2((-1)(x 2)) = a(x 2) + ( 2)(x 2) = a(x 2) 2(x 2) a(x 2) 2(x 2) = (x 2)( )(x 2)(x 2)a2

Ex: Factor b(a 7) 3(7 a)Common factor among the terms? Well, kind of . . . a 7 is close to 7 - aRecall: (-1)(a 7) = - a + 7 = 7 a b(a 7) 3(7 a) = b(a 7) 3((-1)(a 7)) = b(a 7) + 3(a 7) = b(a 7) +3(a 7) b(a 7) + 3(a 7) = (a 7)( + )(a 7)(a 7)b3

Factor by GroupingIf the polynomial has four terms, consider factor by groupingFactor out the GCF from the first two termsFactor out the GCF from the second two terms (take the negative sign if minus separates the first and second groups)If factor by grouping is the correct approach, there should be a common factor among the groupsFactor out that GCFCheck by multiplying using FOIL

Ex: Factor 6a3 + 3a2 +4a + 2Notice 4 terms . . . think two groups: 1st two and 2nd two Common factor among the 1st two terms? 6a3 + 3a2 = 3a2( + )GCF(6a3, 3a2) = 3a2 3a23a22a12a1Common factor among the 2nd two terms? GCF(4a, 2) = 2 4a + 2 = 2( + )22212a1Now put it all together . . .

6a3 + 3a2 +4a + 2 = 3a2(2a + 1) + 2(2a + 1)Four terms two terms. Is there a common factor?Each term has factor (2a + 1)3a2(2a + 1) + 2(2a + 1)= (2a + 1)( + )(2a + 1)(2a + 1)3a226a3 + 3a2 +4a + 2 = (2a + 1)(3a2 + 2)

Ex: Factor 4x2 + 3xy 12y 16x Notice 4 terms . . . think two groups: 1st two and 2nd two Common factor among the 1st two terms? 4x2 + 3xy = x( + )GCF(4x2, 3xy) = x xx4x3y4x3yCommon factor among the 2nd two terms? GCF(-12y, - 16x) = -4 -12y 16x = - 4( )-4-43y4x3y+ 4xNow put it all together . . .

4x2 + 3xy 12y 16x = x(4x + 3y) 4(4x + 3y)Four terms two terms. Is there a common factor?Each term has factor (4x + 3y)x(4x + 3y) 4(4x + 3y)= (4x + 3y)( )(4x + 3y)(4x + 3y)x 4 4x2 + 3xy 12y 16x = (4x + 3y)(x 4)

Ex: Factor 2ra + a2 2r a Notice 4 terms . . . think two groups: 1st two and 2nd two Common factor among the 1st two terms? 2ra + a2 = a( + )GCF(2ra, a2) = a aa2raCommon factor among the 2nd two terms? GCF(-2r, - a) = -1 -2r a = - 1( )-1-12r+ aNow put it all together . . .

2ra + a2 2r a = a(2r + a) 1(2r + a)Four terms two terms. Is there a common factor?Each term has factor (2r + a)a(2r + a) 1(2r + a)= (2r + a)( )(2r + a)(2r + a)a 1 2ra + a2 2r a = (2r + a)(a 1)

To factor a polynomial completely, askDo the terms have a common factor (GCF)?Does the polynomial have four terms? Is the polynomial a special one?Is the polynomial a difference of squares?a2 b2Is the trinomial a perfect-square trinomial?a2 + 2ab + b2 or a2 2ab + b2Is the trinomial a product of two binomials?Factored completely?

Special PolynomialsIs the polynomial a difference of squares?a2 b2 = (a b)(a + b)

Is the trinomial a perfect-square trinomial?a2 + 2ab + b2 = (a + b)2a2 2ab + b2 = (a b)2

Ex: Factor x2 4 Notice the terms are both perfect squaresx2 = (x)24 = (2)2 x2 4 = (x)2 (2)2a2 b2and we have a difference= (x 2)(x + 2) difference of squares= (a b)(a + b)factors as

Ex: Factor 9p2 16 Notice the terms are both perfect squares9p2 = (3p)216 = (4)2 9a2 16 = (3p)2 (4)2a2 b2and we have a difference= (3p 4)(3p + 4) difference of squares= (a b)(a + b)factors as

Ex: Factor y6 25 Notice the terms are both perfect squaresy6 = (y3)225 = (5)2 y6 25 = (y3)2 (5)2a2 b2and we have a difference= (y3 5)(y3 + 5) difference of squares= (a b)(a + b)factors as

Ex: Factor 81 x2y2 Notice the terms are both perfect squares81 = (9)2x2y2 = (xy)2 81 x2y2 = (9)2 (xy)2a2 b2and we have a difference= (9 xy)(9 + xy) difference of squares= (a b)(a + b)factors as

FOIL Method of FactoringRecall FOIL (3x + 4)(4x + 5) = 12x2 + 15x + 16x + 20 = 12x2 + 31x + 20

The product of the two binomials is a trinomialThe constant term is the product of the L termsThe coefficient of x, b, is the sum of the O & I productsThe coefficient of x2, a, is the product of the F terms

FOIL Method of FactoringFactor out the GCF, if anyFor the remaining trinomial, find the F terms (__ x + )(__ x + ) = ax2Find the L terms ( x + __ )( x + __ ) = cLook for the outer and inner products to sum to bxCheck the factorization by using FOIL to multiply

Ex: Factor b2 + 6b + 51. there is no GCF2. the lead coefficient is 1 (1b )(1b ) 3. Look for factors of 5 1, 5 & 5, 1 (b + 1)(b + 5) or (b + 5)(b + 1) 4. outer-inner product?(b + 1)(b + 5) 5b + b = 6b or (b + 5)(b + 1) b + 5b = 6b Either one works b2 + 6b + 5 = (b + 1)(b + 5)5. check: (b + 1)(b + 5) =b2 + 5b + b + 5= b2 + 6b + 5

Ex: Factor y2 + 6y 55 1. there is no GCF2. the lead coefficient is 1 (1y )(1y ) 3. Look for factors of 55 1, -55 & 5, - 11 & 11, - 5 & 55, - 1(y + 1)(y 55) or (y + 5)(y - 11) or ( y + 11)(y 5) or (y + 55)(y 1) 4. outer-inner product?(y + 1)(y - 55) -55y + y = - 54y (y + 55)(y - 1) -y + 55y = 54y y2 + 6y - 55 = (y + 11)(y 5)5. check: (y + 11)(y 5) =y2 5y + 11y - 55= y2 + 6y 55 (y + 5)(y - 11) -11y + 5y = -6y (y + 11)(y - 5) -5y + 11y = 6y

Factor completely 3 TermsAlways look for a common factorimmediately take it out to the front of the expression all common factorsshow whats left inside ONE set of parenthesisIdentify the number of terms. If there are three terms, and the leading coefficient is positive:find all the factors of the first term, find all the factors of the last termWithin 2 sets of parentheses, place the factors from the first term in the front of the parenthesesplace the factors from the last term in the back of the parenthesesNEVER put common factors together in one parenthesis.check the last sign, if the sign is plus: use the SAME signs, the sign of the 2nd termif the sign is minus: use different signs, one plus and one minussmile to make sure you get the middle termmultiply the inner most terms together then multiply the outer most terms together, and add the two products together.

Factor completely: 2x2 5x 7 Factors of the first term: 1x & 2x

Factors of the last term: -1 & 7 or 1 & -7

(2x 7)(x + 1)

Factor Completely.

4x2 + 83x + 60Nothing commonFactors of the first term: 1 & 4 or 2 & 2Factors of the last term: 1,6 2,30 3,20 4,15 5,12 6,10Since each pair of factors of the last has an even number, we can not use 2 & 2 from the first term (4x + 3)(1x + 20 )

Sign Pattern for the BinomialsTrinomial Sign PatternBinomial Sign Pattern + + ( + )( + ) - + ( - )( - ) - - 1 plus and 1 minus + - 1 plus and 1 minusBut as you can tell from the previous example, the FOIL method of factoring requires a lot of trial and error (and hence luck!) . . . Better way?

ac Method for factoring ax2 + bx + cFactor out the GCF, if anyFor the remaining trinomial, multiply acLook for factors of ac that sum to bRewrite the bx term as a sum using the factors found in step 3Factor by groupingCheck by multiplying using FOIL

Ex: Factor 3x2 4x 15 1. Is there a GCF? No2. Multiply ac a = 33and c = 15 15 3(-15) = - 45 3. Factors of -45 that sum to 4 4 4. Rewrite the middle term 3x2 4x 15 = 3x2 9x + 5x 15 1 45 44 3 15 12 5 9 4 Note: although there are more factors of 45, we dont have to check them since we found what we were looking for!Four-term polynomial . . . How should we proceed to factor?

Factor by grouping . . . 3x2 9x + 5x 15Common factor among the 1st two terms? 3 x 2 9x = 3x( ) 3x 3x3x3x3Common factor among the 2nd two terms? 5 5 x 15 = 5( )553x3 3x2 9x + 5x 15 = 3x(x 3) + 5(x 3) = (x 3)( ) 3x + 5

Ex: Factor 2t2 + 5t 12 1. Is there a GCF? No2. Multiply ac a = 22and c = 12 12 2(-12) = - 24 3. Factors of -24 that sum to + 5 5 4. Rewrite the middle term 2t2 + 5t 12 = 2t2 3t + 8t 12 1 24 23 2 12 10 3 8 5 Four-term polynomial . . . Factor by grouping . . .Close but wrong sign so reverse it- 3 8 5

2t2 3t + 8t 12Common factor among the 1st two terms? 2 t 2 3t = t( ) t tt32t3Common factor among the 2nd two terms? 4 8 t 12 = 4( )4432t3 2t2 3t + 8t 12 = t(2t 3) + 4(2t 3) = (2t 3)( ) t + 42

Ex: Factor 9x4 + 18x2 + 8 1. Is there a GCF? No2. Multiply ac a = 99and c = 8 8 9(8) = 72 3. Factors of 72 that sum to 18 18 4. Rewrite the middle term 9x4 + 18x2 + 8 = 9x4 + 6x2 + 12x2 + 8 1 72 73 3 24 27 6 12 18 Four-term polynomial . . . Factor by grouping . . .Bit big think bigger factors

9x4 + 6x2 + 12x2 + 8Common factor among the 1st two terms? 9x4 + 6x2 = 3x2( + ) 3x2 3x23x223x22Common factor among the 2nd two terms? 4 12x2 + 8 = 4( + )4433x22 9x4 + 6x2 + 12x2 + 8 = 3x2(3x2 + 2) + 4(3x2 + 2) = (3x2 + 2)( + ) 3x2 433x2

Ex: Factor 12x2 17xy + 6y2 1. Is there a GCF? No, but notice two variables2. Multiply ac a = 12x212and c = 6y2 6y2 12x2(6y2) = 72y2 3. Factors of 72x2y2 that sum to 17 y - 17xy4. Rewrite the middle term 12x2 17xy + 6y2 = 12x2 8xy 9xy + 6y2-1xy -72xy -73xy -6xy -12xy -18xy -8xy -9xy -17xy Four-term polynomial . . . Factor by grouping . . .Each factor need a y, both need to be negativeToo big, think bigger factorsPick one to be the variable

12x2 8xy 9xy + 6y2 Common factor among the 1st two terms? 12x2 8xy = 4x( ) 4x 4x4x2y3x2yCommon factor among the 2nd two terms? - 3y 9xy + 6y2 = - 3y( )-3y-3y-2y3x 2y 12x2 8xy 9xy + 6y2 = 4x(3x 2y) 3y(3x 2y) = (3x 2)( ) 4x 3y33x

Ex: Factor x3 + 3x2 4x 12 1. Is there a GCF? No2. Notice four terms grouping Common factor among the 1st two terms? x2 x3 + 3x2 = x2( + ) x2 x2 x x 3Common factor among the 2nd two terms? - 4 4x 12 = 4( ) - 4 - 4 3 x+ 3 x3 + 3x2 - 4x 12 = x2(x + 3) 4(x + 3) = (x + 3)( ) x2 4

Cont: we have (x + 3)(x2 4)But are we done? No. We have to make sure we factor completely.Is (x + 3) prime? can x + 3 be factored further? No . . . It is primeWhat about (x2 4)? Recognize it?Difference of Squaresx2 = (x)24 = (2)2 x2 4 = (x)2 (2)2= (x 2)(x + 2)Therefore x3 + 3x2 4x 12 = (x + 3)(x2 4) = (x + 3)(x 2)(x + 2)

Special PolynomialsIs the polynomial a sum/difference of cubes?a3 + b3 = (a + b)(a2 - ab + b2)a3 b3 = (a - b)(a2 + ab + b2)

Ex: Factor 8p3 q3 Notice the terms are both perfect cubes8p3 = (2p)3q3 = (q)3 8p3 q3 = (2p)3 (q)3a3 b3and we have a difference= (2p q)((2p)2 + (2p)(q) + (q)2) difference of cubes= (a b)(a2 + ab + b2)factors as= (2p q)(4p2 + 2pq + q2)

Ex: Factor x3 + 27y9 Notice the terms are both perfect cubesx3 = (x)327y9 = (3y3)3 x3 + 27y9 = (x)3 + (3y3)3a3 + b3and we have a sum= (x + 3y3)((x)2 - (x)(3y3) + (3y3)2) sum of cubes= (a + b)(a2 - ab + b2)factors as= (x + 3y3)(x2 3xy3 + 9y6)

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INTRODUCTION TO FACTORING POLYNOMIALS