1 Factoring Methods When factoring a polynomial, there are three steps you want to follow. 1. Check if thereβs a Greatest Common Factor (GCF) 2. Check if the polynomial can be factored by grouping (4 term polynomial) 3. Check if the polynomial is a trinomial or a Special Case Greatest Common Factor A Common Factor is a number that can be divided into each term of the polynomial evenly. The Greatest Common Factor (G.C.F.) is the largest common factor by which you can evenly divide each term of the polynomial. Keep in mind that 1 is not a GCF, but it might be needed in certain examples when factoring by grouping. Example: Factor 3x2 + 15x The G.C.F. is 3x 1. Write the G.C.F. before of the parentheses 2. Divide both terms by the G.C.F. 3. Write the quotient inside of the parentheses 3π₯ 2 15π₯ + 3π₯ 3π₯ 3x(x + 5) Other examples: Factor m2 + mn 12x3 + 16x2 β 8x SLC Lake Worth Math Lab m(m + n) 4x(3x2 + 4x β 2) Factoring 2 Factoring by Grouping This factoring method applies to four term polynomials. The steps are as follows: 1. Check if thereβs a G.C.F. 2. If thereβs no G.C.F. factor as follows: Example: Factor 2ab + 12a + 3b + 18 2ab + 12a|+ 3b + 18 Make two groups of two terms and take G.C.F. of each pair separately 2a(b + 6) + 3(b + 6) Factor (b + 6) which is the GCF from each group (b + 6) (b + 6) (b + 6)(2a + 3) and divide each term by GCF Group the remaining terms in another parentheses Example: (Intermediate Algebra) Factor 6xy β 5 β 15x +2y 6xy - 5|-15x + 2y 1(6xy β 5) β 1(15 -2y) The parentheses donβt match 6xy +2y|β 15x -5 Re-arrange the terms 2y(3x + 1) -5(3x +1) (3x + 1)(2y β 5) Other examples: Factor by Grouping 5q2 β 4pq β 5q +4p (5q β 4p)(q β 1) 16x2 + 4xy2 + 8xy +2y3 2(4x + y2)(2x + y) Intermediate Algebra Examples 4 + xy -2y β 2x (2 β y)(2 β x) 3m2n3 + 15m2n β 2m2n2 β 10m2 m2 (n2 + 5)(3n β 2) SLC Lake Worth Math Lab Factoring 3 Factoring Trinomial of the Form ax2 +bx +c where a = 1 To factor trinomials of this form, the trinomial has to be in descending order. 1. Find factors of c, where the sum of the factors adds up to b Example: Factor x2 - 5x +6 a=1 b=-5 c=6 Find factors of 6 that have a sum of -5 Factors 6 6·1 3·2 Sum of factors -5 -6-1 -3-2 The factors of x2 - 5x +6 are (x - 3)(x β 2) Example: Factor x2 β x β 2 Find factors of -2 that have a sum of -1 Factors -2 2·1 Sum of factors -1 -2+1 (x β 2)(x + 1) Other examples: Factor a2 β 9ab + 18b2 (a β 3b)(a β 6b) 10t β 24 + t2 (t β 2)(t + 12) 3x2 +9x β 30 3(x + 5)(x -2) 6x3 + 54x2 + 120x 6x(x + 4)(x + 5) SLC Lake Worth Math Lab Factoring 4 Factoring Trinomial of the form ax2 +bx +c where a οΎ 1 Determining Signs when Factoring Trinomial of the form ax2 +bx +c where aοΎ1 This is a helpful hint to decide which signs should be placed on each factor when dealing with trinomials of the form ax2 + bx + c: ο· When the last sign is (+), both factors take the sign of the middle term ax2 + bx + c (___ + ___)(___ + ___) ax2 β bx + c (___ - ___)(___ - ___) ο· When the last sing is (-), one factor is (+) and the other is (-) ax2 β bx β c (___ + ___)(___ - ___) ax2 + bx - c (___ + ___)(___ - ___) SLC Lake Worth Math Lab Factoring 5 Scissors Method Example: Factor 3x2 + 10x β 8 1. Check if the trinomial has a G.C.F. 2. The trinomial has to be in descending order and the leading coefficient should be positive. 3. List all the factors of a and c 4. Keep in mind that if the last term of the trinomial is negative, the factors should be one (+) and one (-). 5. List the factors of a and c vertically and multiply crisscross (scissors) as follows: Factors of 3 3·1 Factors of 8 8·1 4·2 (3 2) (1 4) -2 1 times 2 negative sign +12 3 times 4 positive sign 10 (3x - 2)(x + 4) Factor 24x2 + 41x + 12 Keep in mind that the last term of the trinomial is positive, so the factors should have the sign of the middle term. Factors of 24 24·1 12·2 8·3 6·4 Factors of 12 12·1 6·2 4·3 (6 3) 12 (4 4) 24 SLC Lake Worth Math Lab Factoring 6 You can clearly that this combination wonβt work, because there is a G.C.F. for (4 and 4) (8 3) +9 Positive sign (3 4) +32 Positive sign 41 (8x + 3)(3x + 4) Factor 8x2 β 14xy + 3y2 Factors of 8 8·1 4·2 Factors of 3 3·1 (4 3) -6 (2 1) -4 -10 Switch one of the factors (4 1) -2 Negative sign (2 3) -12 Negative sign -14 (4x β 1y)(2x β 3y) Other examples: Factor 25n2 β 5n - 6 (5n + 2)(5n β 3) 12x3 β 34x2 + 24x 2x(3x - 4)(2x β 3) SLC Lake Worth Math Lab Factoring 7 Factoring Trinomial of the form ax2 +bx +c where a οΎ 1 Trial and Error Method Example: Factor 12x2 - 5x β 2 1. Check if the trinomial has a G.C.F. 2. The trinomial has to be in descending order and the leading coefficient should be positive. 3. Find products of a and c. 4. Choose inner and outer terms. Use various combinations of the factors from Step 3 until the necessary middle term is found. (____x ____) (____x ____) Factors of 12 are 12·1 6·2 4·3 We try 4·3 for the first two terms ( 4x )( 3x ) Factor of -2 are -2·1 or 2·-1. We try both possibilities (4x - 2 )(3x + 1) Notice that 4x β 2 have a common factor, so this combination is wrong. 8x (4x - 1 )(3x + 2) 8x β 3x = 5x. Wrong middle term -3x Interchange the sings SLC Lake Worth Math Lab Factoring 8 -8x (4x + 1 )(3x - 2) -8x + 3x = -5x. Correct middle term 3x Therefore the factors of 12x2 - 5x β 2 are (4x + 1 )(3x - 2) Other examples: 18m2 β 19mn β 12n2 (9m + 4n)(2m β 3n) -3x2 + 16x + 12 -(3x + 2)(x β 6) SLC Lake Worth Math Lab Factoring 9 Special cases Perfect Square Trinomials A perfect square trinomial has the following form: a2 + 2ab + b2 = (a + b)2 a2 β 2ab + b2 = (a β b)2 To identify a perfect square trinomial, check the following: 1. The first term and the last term are perfect squares. 2. The coefficient of the middle term is twice the square root of the first term multiplied by the square root of the last term. 3. The last sign has to be positive. Example: Factor 4m2 + 20m + 25 (2m)2 + (5)2 Middle term 2(2m)(5) is 20m 4m2 + 20m + 25 is a Perfect Square Trinomial. The factored form is: (2m + 5)2 Example: Factor p2 - 8p + 64 (p)2 + (8)2 Middle term 2(p)(8) is not 8p Not a Perfect Square Trinomial Other Examples: Factor 144a2 β 216ab + 81b2 (12a - 9b)2 49x2 + 14xy + y2 (7x + y)2 SLC Lake Worth Math Lab Factoring 10 Difference of Perfect Squares The Difference of Perfect Squares has the following form: a2 β b2 = (a + b)(a β b) To identify a difference of perfect squares 1. The first term and the last term are perfect squares 2. The terms have opposite signs Example: Factor x2 β 16 Both terms are perfect squares and they have opposite signs (x + 4)(x -4) Example: Factor 81a2 β 25b2 Both terms are perfect squares and they have opposite signs (9a + 5b)(9a β 5b) Other examples: 4x2 β 25y2 (2x + 5y)(2x β 5y) 36a2b2 β 1 (6ab + 1)(6ab β 1) 2y2 β 18 2(y + 3)(y β 3) 3x3 β 48x 3x(x + 4)(x β 4) Intermediate Algebra Examples: a4 - 16 SLC Lake Worth Math Lab (a2 + 4)(a + 2)(a β 2) Factoring 11 Sum and Difference of Perfect Cubes (Intermediate Algebra) The Sum and Difference of perfect Cubes have the following general forms: a3 β b3 = (a - b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 - ab + b2) To identify a difference of perfect squares 1. Both terms have to be perfect cubes 2. Follow the general form Example: Factor x3 + 27 (x + 3)(x2 β 3x + 32) (x + 3)(x2 β 3x + 9) Example: Factor 8x3 β 125 (2x β 5)[(2x)2 + 10x + (5)2] (2x β 5)(2x2 + 10x + 25) Other examples 8p3 -1 (2p β 1)( 4p2 + 2p +1) 2x3 + 2000 2(x + 10)(x2 β 10x + 100) SLC Lake Worth Math Lab Factoring 12 Factoring a Polynomial Using Substitution (Intermediate Algebra) Example: Factor (y + z)2 - 81 Let t = y + z t2 β 81 Substitute t in the original binomial (t + 9)(t β 9) Factor the difference of perfect squares (y + z + 9)(y + z β 9) Replace t with y + z Example: Factor 2(x + 3)2 + 5(x + 3) β 12 Let t = x + 3 2t2 + 5t β 12 Factor using scissors or trial and error (2t β 3)(t + 4) Replace t with x + 3 [2(x + 3) β 3][(x + 3) + 4] (2x + 6 β 3)(x + 3 +4) Simplify (2x + 3)(x + 7) Example: Factor (a β 4)3 β b3 Let t = a β 4 t3 β b3 Factor as difference of cubes (t β b)(t2 + tb + b2) Replace t with a β 4 (a β 4 β b)[(a β 4)2 + (a β 4)b + b2] (a β 4 β b)(a2 β 8a + 16 β ab β 4b + b2) Other examples: (x β 2y)2 β 4 (x β 2y + 2)(x β 2y -2) 4(p + 2) + m(p + 2) (p + 2)(4 + m) (p + 8q)2 β 10(p + 8q) + 25 (p + 8q β 5)2 SLC Lake Worth Math Lab Factoring

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