5.4 Factoring Polynomials Objectives: 1.Factor polynomials 2.Simplify polynomial quotients by factoring.

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Presentation transcript:

5.4 Factoring Polynomials Objectives: 1.Factor polynomials 2.Simplify polynomial quotients by factoring.

Factoring 1.Look for the greatest common factor (GCF) if there is one. Factor it out. 2.Count the number of terms remaining after taking out the GCF. There are limited ways to factor for each number of terms. See chart on next slide. 3.Factor. Make sure your answer does not contain any common factors other than one. Some polynomials may not factor. Write “prime” if this occurs.

Counting Terms 1.Two terms  Difference of Squaresa²-b²=(a-b)(a+b)  Sum of Two Cubesa³+b³=(a+b)(a²-ab+b²)  Difference of 2 Cubesa³-b³=(a-b)(a²+ab+b²) 2.Three Terms  Perfect Square Trinomialsa²+2ab+b²=(a+b)² or a²-2ab+b²=(a-b)²  General Trinomials like x²-2x+8 = (x-4)(x+2) 3.4 or more  Grouping

GCF To factor the GCF out of a polynomial, divide it into each term. Example:5x⁴y³-15x³y²z GCF: 5x³y² Answer: 5x³y²(xy-3z)

Difference of Squares and Cubes These are easy to factor if you know what to look for. Just learn the pattern, how to recognize it and “fill in the blanks” Examples:x²-16y² (all are perfect squares and it is a difference) (x-4y)(x+4y) For sums and differences of cubes, it might help to know a few numerical cubes. 1, 8, 27, 64, 125… (variables must be multiples of 3) x³y³+8(xy+2)(x²y²-2xy+4)

Grouping Pair up terms so that a GCF can be taken out of each of them. After removing the GCF, the parenthesis should be the same. If not, you may have to pair them up differently. If they are the same, rewrite with the GCF’s together and the common parenthesis written once. Example: 21-7x+3y-xy 7(3-x)+y(3-x) (7+y)(3-x)

Trinomials Factor 1.x²-8x+16 2.x²-2x-48 3.x²-15x+50

Examples 1.10a³b²+15a²b-5ab³ 1. 5ab(2a²b+3a-b²) (GCF only) 2.x³+5x²-2x-10 (x³+5x²)-(2x-10) x²(x+5)+(-2)(x+5) (x²-2)(x+5) 3.5xy²-45x 5x(y²-9) 5x(y-3)(y+3) 4.64x⁶-y⁶ (8x³-y³)(8x³+y³) (2x-y)(4x²+2xy+y²)(2x+y) (4x²-2xy+y²)

Simplifying Quotients 1.Factor 2.Cancel any like-binomials. Do not cancel single terms that are being added or subtracted to or from another term. Example: a²-a-6(a-3)(a+2) a²+7a+10(a+2)(a+5) a-3 a+5

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