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INTRODUCTION TO FACTORING POLYNOMIALS

INTRODUCTION TO FACTORING POLYNOMIALS. MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur. Definitions. Recall: Factors of a number are the numbers that divide the original number evenly.

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INTRODUCTION TO FACTORING POLYNOMIALS

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  1. INTRODUCTION TO FACTORING POLYNOMIALS MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur

  2. Definitions • Recall: 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.

  3. Ex: Find the GCF(24, 40). Prime factor each number: 24 2 12 2 6 2 3  24 = 2*2*2*3 = 23*3  GCF(24,40) = 23 = 8 40 2 20 2 10 2 5  40 = 2*2*2*5 = 23*5

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

  5. Ex: Find the GCF(6a5,3a3,2a2) • 6a5 = 2*3*a*a*a*a*a • 3a3 = 3*a*a*a • 2a2 = 2*a*a The terms share two factors of a  GCF(6a5,3a3,2a2)=a2 Note: The exponent of the variable in the GCF is the smallest exponent of that variable the terms

  6. Definitions • To factor an expression means to write an equivalent expression that is a product • To factor a polynomial means to write the polynomial as a product of other polynomials • A 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

  7. To factor a polynomial completely, ask • Do 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 – b2 • Is the polynomial a sum/difference of cubes? • a3 + b3 or a3 – b3 • Is the trinomial a perfect-square trinomial? • a2 + 2ab + b2 or a2 – 2ab + b2 • Is the trinomial a product of two binomials? • Factored completely?

  8. Ex: Factor 7x2 + 3x Think 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) = x Factor out 2  7 x + 3 x = x ( + ) What’s left? x x  7x2 + 3x = x(7x + 3)

  9. Ex: Factor 6a5 – 3a3 – 2a2 Recall: GCF(6a5,3a3,2a2)=a2 3 1 6a5 – 3a3 – 2a2 =a2( - - ) 6a3 3a 2 a2 a2 a2  6a5 – 3a3 – 2a2 = a2(6a3 – 3a – 2)

  10. Your Turn to Try a Few

  11. 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)( – ) x 2 (a + b) (a + b)  x(a + b) – 2(a + b) = (a + b)(x – 2)

  12. 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)( – ) a 2 (x – 2) (x – 2)

  13. Ex: Factor b(a – 7) – 3(7 – a) Common factor among the terms? Well, kind of . . . a – 7 is close to 7 - a Recall: (-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)( + ) b 3 (a – 7) (a – 7)

  14. Your Turn to Try a Few

  15. To factor a polynomial completely, ask • Do 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 – b2 • Is the polynomial a sum/difference of cubes? • a3 + b3 or a3 – b3 • Is the trinomial a perfect-square trinomial? • a2 + 2ab + b2 or a2 – 2ab + b2 • Is the trinomial a product of two binomials? • Factored completely?

  16. Factor by Grouping • If the polynomial has four terms, consider factor by grouping • Factor out the GCF from the first two terms • Factor 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 groups • Factor out that GCF • Check by multiplying using FOIL

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

  18. 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) = (2a + 1)( + ) 3a2 2 3a2(2a + 1) + 2(2a + 1) (2a + 1) (2a + 1) 6a3 + 3a2 +4a + 2 = (2a + 1)(3a2 + 2)

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

  20. 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) = (4x + 3y)( ) x – 4 x(4x + 3y) – 4(4x + 3y) (4x + 3y) (4x + 3y) 4x2 + 3xy – 12y – 16x = (4x + 3y)(x – 4)

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

  22. 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) = (2r + a)( ) a – 1 a(2r + a) – 1(2r + a) (2r + a) (2r + a) 2ra + a2 – 2r – a = (2r + a)(a – 1)

  23. Your Turn to Try a Few

  24. To factor a polynomial completely, ask • Do 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 – b2 • Is the trinomial a perfect-square trinomial? • a2 + 2ab + b2 or a2 – 2ab + b2 • Is the trinomial a product of two binomials? • Factored completely?

  25. Special Polynomials • Is the polynomial a difference of squares? • a2 – b2 = (a – b)(a + b) • Is the trinomial a perfect-square trinomial? • a2 + 2ab + b2 = (a + b)2 • a2 – 2ab + b2 = (a – b)2

  26. Ex: Factor x2 – 4 Notice the terms are both perfect squares  difference of squares and we have a difference x2 = (x)2 4 = (2)2  x2 – 4 = (x)2 – (2)2 = (x – 2)(x + 2) a2 – b2 = (a – b)(a + b) factors as

  27. Ex: Factor 9p2 – 16 Notice the terms are both perfect squares  difference of squares and we have a difference 9p2 = (3p)2 16 = (4)2  9a2 – 16 = (3p)2 – (4)2 = (3p – 4)(3p + 4) a2 – b2 = (a – b)(a + b) factors as

  28. Ex: Factor y6 – 25 Notice the terms are both perfect squares  difference of squares and we have a difference y6 = (y3)2 25 = (5)2  y6 – 25 = (y3)2 – (5)2 = (y3 – 5)(y3 + 5) a2 – b2 = (a – b)(a + b) factors as

  29. Ex: Factor 81 – x2y2 Notice the terms are both perfect squares  difference of squares and we have a difference 81 = (9)2 x2y2 = (xy)2  81 – x2y2 = (9)2 – (xy)2 = (9 – xy)(9 + xy) a2 – b2 = (a – b)(a + b) factors as

  30. Your Turn to Try a Few

  31. To factor a polynomial completely, ask • Do 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 – b2 • Is the polynomial a sum/difference of cubes? • a3 + b3 or a3 – b3 • Is the trinomial a perfect-square trinomial? • a2 + 2ab + b2 or a2 – 2ab + b2 • Is the trinomial a product of two binomials? • Factored completely?

  32. FOIL Method of Factoring • Recall FOIL • (3x + 4)(4x + 5) = 12x2 + 15x + 16x + 20 = 12x2 + 31x + 20 • The product of the two binomials is a trinomial • The constant term is the product of the L terms • The coefficient of x, b, is the sum of the O & I products • The coefficient of x2, a, is the product of the F terms

  33. FOIL Method of Factoring • Factor out the GCF, if any • For the remaining trinomial, find the F terms (__ x + )(__ x + ) = ax2 • Find the L terms ( x + __ )( x + __ ) = c • Look for the outer and inner products to sum to bx • Check the factorization by using FOIL to multiply

  34. Ex: Factor b2 + 6b + 5 1. there is no GCF 2. 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-innerproduct? (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

  35. Ex: Factor y2 + 6y – 55 1. there is no GCF 2. 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-innerproduct? (y + 1)(y - 55)  -55y + y = - 54y (y + 5)(y - 11)  -11y + 5y = -6y (y + 55)(y - 1)  -y + 55y = 54y (y + 11)(y - 5)  -5y + 11y = 6y  y2 + 6y - 55 = (y + 11)(y – 5) 5. check: (y + 11)(y – 5) = y2 – 5y + 11y - 55 = y2 + 6y – 55

  36. Factor completely – 3 Terms Always look for a common factor immediately take it out to the front of the expression all common factors show what’s left inside ONE set of parenthesis Identify 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 term Within 2 sets of parentheses, place the factors from the first term in the front of the parentheses place the factors from the last term in the back of the parentheses NEVER 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 term if the sign is minus: use different signs, one plus and one minus “smile” to make sure you get the middle term multiply the inner most terms together then multiply the outer most terms together, and add the two products together.

  37. 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)

  38. Factor Completely.4x2 + 83x + 60 Nothing common Factors of the first term: 1 & 4 or 2 & 2 Factors of the last term: 1,6 2,30 3,20 4,15 5,12 6,10 Since 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 )

  39. Sign Pattern for the Binomials Trinomial Sign Pattern Binomial Sign Pattern + + ( + )( + ) - + ( - )( - ) - - 1 plus and 1 minus + - 1 plus and 1 minus But 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?

  40. Your Turn to Try a Few

  41. ac Method for factoring ax2 + bx + c • Factor out the GCF, if any • For the remaining trinomial, multiply ac • Look for factors of ac that sum to b • Rewrite the bx term as a sum using the factors found in step 3 • Factor by grouping • Check by multiplying using FOIL

  42. – 15 Ex: Factor 3x2 – 4x – 15 3 – 4 No 1. Is there a GCF? 3(-15) = - 45 2. Multiply ac  a = 3 and c = – 15 3. Factors of -45 that sum to – 4 1 – 45  – 44 Note: although there are more factors of – 45, we don’t have to check them since we found what we were looking for! 3 – 15  – 12 5 – 9  – 4 4. Rewrite the middle term 3x2– 4x – 15 = 3x2– 9x + 5x – 15 Four-term polynomial . . . How should we proceed to factor?

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

  44. Ex: Factor 2t2 + 5t – 12 2 + 5 – 12 No 1. Is there a GCF? 2(-12) = - 24 2. Multiply ac  a = 2 and c = – 12 3. Factors of -24 that sum to 5 1 – 24  – 23 Close but wrong sign so reverse it - 3 8  5 2 – 12  – 10 3 – 8  – 5 4. Rewrite the middle term 2t2+ 5t – 12 = 2t2– 3t + 8t – 12 Four-term polynomial . . . Factor by grouping . . .

  45. 2t2 – 3t + 8t – 12 Common factor among the 1st two terms? t 3 2t  2 t 2 – 3t = t( – ) 3 t t Common factor among the 2nd two terms? 4 2 3  8 t – 12 = 4( – ) 2t 3 4 4  2t2 – 3t + 8t – 12 = t(2t – 3) + 4(2t – 3) = (2t – 3)( ) t + 4

  46. Ex: Factor 9x4 + 18x2 + 8 9 18 8 No 1. Is there a GCF? 9(8) = 72 2. Multiply ac  a = 9 and c = 8 3. Factors of 72 that sum to 18 1 72  73 Bit big  think bigger factors 3 24  27 6 12  18  4. Rewrite the middle term 9x4+ 18x2 + 8 = 9x4+ 6x2 + 12x2 + 8 Four-term polynomial . . . Factor by grouping . . .

  47. 9x4 + 6x2 + 12x2 + 8 Common factor among the 1st two terms? 3x2 2 3x2 3x2  9x4 + 6x2 = 3x2( + ) 2 3x2 3x2 Common factor among the 2nd two terms? 4 3 3  12x2 + 8 = 4( + ) 3x2 2 4 4  9x4 + 6x2 + 12x2 + 8 = 3x2(3x2 + 2) + 4(3x2 + 2) = (3x2 + 2)( + ) 3x2 4

  48. – 17y Ex: Factor 12x2 – 17xy + 6y2 12 6y2 Pick one to be the variable No, but notice two variables 1. Is there a GCF? 12x2(6y2) = 72y2 2. Multiply ac  a = 12x2 and c = 6y2 3. Factors of 72x2y2 that sum to - 17xy -1xy -72xy  -73xy Each factor need a y, both need to be negative -6xy -12xy  -18xy Too big, think bigger factors -8xy -9xy  -17xy  4. Rewrite the middle term 12x2– 17xy + 6y2 = 12x2– 8xy – 9xy + 6y2 Four-term polynomial . . . Factor by grouping . . .

  49. 12x2 – 8xy – 9xy + 6y2 Common factor among the 1st two terms? 4x 2y 3x 3x  12x2 – 8xy = 4x( – ) 2y 4x 4x Common factor among the 2nd two terms? - 3y 3 -2y  – 9xy + 6y2= - 3y( ) 3x – 2y -3y -3y  12x2 – 8xy – 9xy + 6y2 = 4x(3x – 2y) – 3y(3x – 2y) = (3x – 2)( ) 4x – 3y

  50. Your Turn to Try a Few

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