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Polynomials & Factoring (A-5) Polynomial: the most common type of algebraic expression. Monomial: an algebraic expression consisting of a single term. Binomial: an algebraic expression that has two terms. Trinomial:. an algebraic expression that has three terms. Common factor: a whole number that is a factor of each number in a set of numbers. Factor: a number or expression that is multiplied by another to yield a product. Remember that factoring is just "un-doing" multiplication. Take the number 6, for example. We can write it as factors like this: (2)(3). It works the same way with polynomials, but we have to consider our variables and GCF. Greatest common factor (GCE):the greatest number that is a factor of two or more numbers Identify the GCF(greatest common factor) in each expression. 1.4x+2y 2. 8x2 +9x x Quadratic equation: a polynomial equation of the second degree, generally expressed as ax2 c·= 0, where a, band c are real numbers and a is not equal to zero. Types of Factoring to consider: Take out GCF(check this first every time you factor) Difference of squares Sum & difference of cubes Perfect square trinomial Trinomial with leading coefficient of 1 Trinomial with leading coefficient NOT 1 Grouping ALWAYS FIRST: look for a GCFin all terms. If there is one, factor out the GCFin front The parentheses should contain the "left over" numbers after taking out the GCF. Example: 15xy2 - 10 x3y + 25 xy3 Factored completely as 5xy( 3y - 2X2+ 5y2) + bx + IF THERE ARE 2 TERMS: it could be ..... Difference of perfect squares: Your answer is the product of two binomials. Take the square root of each term. One binomial will have a positive sign between the terms and the other binomial will have a negative between the terms. Example: 4X2- 2Sy4 factors to (2x + Sy2)(2x - Sy2) Sum or Difference of a perfect cube: Your answer is a binomial times a trinomial. Binomial part is the cube root of each original term with the same sign. For the Trinomial part only look at the new binomial (not the original problem and use SOFAS). Square the first term, Opposite sign, multiply the first term by the second term, Always positive, Square the second term. Example: 8x3 - y3 factors to (2x - y)(4X2+ 2xy + y2) IF THERE ARE 3 TERMS: it could be ..... Perfect square trinomial: If the first and last term are perfect squares and the middle term is twice the product of the" square root of the first and last. Example: X2+ 6x + 9 factors to (x+3)(x+3) which can be written as (x + 3)2 Trinomial with leading coefficient of 1: You must find two numbers that multiply to equal the last term and add to equal the middle term, Your answer is the product of two binomials. Example: "x2 - 4x - 12 find numbers that multiply to -12 and add to -4. The numbers are-6 and + 2, so the final factors are (x + 2)(x - 6) Trinomial with leading coefficient not equal to 1: Your answer will be the product of two binomials. There are a few different methods that teachers often use. "The Magic Box","T-chart", guess and check. I will do the divide, reduce and swing method here. Example: 3x2 + lOx + 8 Swing the 3 to multiply to the 8 in order to obtain x2 + lOx + 24 Two numbers that multiply to be 24 and add to be 10 are 6 and 4, so we get the factors (x+6)(x+4) Now you must divide each constant by the 3 that was multiplied to the constant in the 6 4 3 3 first step to get (x + - )(x + -) If possible, reduce all fractions to get ex + 2) ex + ~) 3 Now, swing all denominators (that are not one) in front of the variable within the factor to get (3x + 4) (x + 2). IF THERE ARE 4 TERMS: Grouping. Group two terms together and another two terms together leaving an addition sign in between the two binomials. Take out a GCF from each binomial. Your goal is that inside the parentheses are identical. If this happens then you rewrite your answer as the product of two binomials. Example: 12ac + 21ad + 8bc +14bd Step 1. (12ac + 21ad) + (8bc + 14bd) Step 2. 3a(4c + 7d) + 2b(4c + 7d) Step 3. (3a + 2b)(4c + 7d) Here are some more examples: 16m2n + 12mn2 GCF \... X GCF ~ (X -2-) ~ff. of squares Diff. of squares (X +5) (X - 5) ()<'Z t l) ()<"2-_ l) -7 ~ [(X"-tIJ{xtl)(J(-I) GCF! diff. of squares v, x3 + 125 ) > 3 (a -C1b?\? (tA 1- 3 b)[iA-3bJ\ Sum of cubes (!. + 5) ( x-z. - 5X +~5) Z ~ Diff. of cubes (0.. - Sumofcubes ('/. '3- ( '1.- LIZ l-,~\ b) (} +- tl0 + () ) +3jCX 'l- -8X -I- q ) ~ I x2 + 8x + 16 2b3 + 32b2 + 128b (y.. -t- Lf)(x -+- 4 ) Perfect Square Trinomial GCF & trinomial ~ b( b2. 1- 1(.0 l? + I? h [b + ~Xb 4w2 12wr + 9r2 - -r~ (0 If) )J ~ .?b [I?MlY Perfect square trinomial (?W - Br)(?vI - ~r) (z X + Y) (L/ xz..- '2 xy +- y 2. ) Sum of perfect cubes 20x2 - 60x - 35 GCF & Trinomial 5" (IL 'Z '"T)<-J2X-I t; 1)( t I )('2)( -I ) 200 + 17r - r2 Trinomial (~ + 6x2 + lOx - 4 rf?5 - GCF & Trinomial r) \ ,2 (~X~1- 5X -P) \"1- (3)( 4x2 21- + 8x - 4xy - 8y xy - 7y + 3x fR0n:x"p! GCF & Grouping Grouping ~)CPI-'f.y-7y - I)(x -t 1.)\ L/)('" ti1tl/xy -ll0 .. 4x(x +~) -4Y Cx-t-z.) (?c f 2j (4 '1-lJy) -> tEn)(lC- Y) ( ill. -> 3(x+-')-Y (X·FI) -? K>C-I-I)[3:"Y))