Polynomials FOIL and Factoring

 Polynomial – many terms  Standard form – terms are arranged from largest exponent to smallest exponent  Degree of a Polynomial – largest exponent  Leading Coefficient – the first coefficient when written in standard form.  Classification BY NUMBER OF TERMS Monomial : one term Binomial : two terms Trinomial : three terms n-nomial: n terms (more than three terms) BY DEGREE Zero:constant One:linear Two:quadratic Three:cubic Four: quartic n>4:n th degree

 Addition: ignore parentheses and combine like terms. (2x 3 -5x 2 -7x+4) + ( - 6x 3 -2x 2 +x+6) = - 4x 3 -7x 2 -6x+10  Subtraction: distribute the minus to all terms in parentheses behind then combine like terms. (2x 3 -5x 2 -7x+4) - ( - 6x 3 -2x 2 +x+6) = (2x 3 -5x 2 -7x+4) + (- - 6x x x- + 6) = (2x 3 -5x 2 -7x+4) + (6x 3 +2x 2 -x-6) = 8x 3 -3x 2 -8x-2

 General Rule: Multiply every term of one polynomial by every term of the other  Special Polynomial Multiplications: Distributive Property: (Monomial)·(any polynomial) - 3x 2 (5x 2 -6x+2) = (-3x 2 )(5x 2 ) + (-3x 2 )(-6x) + (-3x 2 )(2) = - 15x x 3 - 6x 2 FOIL: (Binomial)·(binomial) First Outer Inner Last (4x – 5)(2x + 7) = (4x)(2x) + (4x)(7) + ( - 5)(2x) + ( - 5)(7) = 8x x – 10x -35 = 8x x -35

(3x + 2) (4x – 5)

Sum and Difference Pattern (a+b)(a–b) = a 2 –ab+ab–b 2 = a 2 –b 2 (3x+5)(3x-5) = (3x) 2 – (5) 2 = 9x Square of a Binomial Pattern (a + b) 2 = (a+b)(a+b) = a 2 +ab+ab+b 2 = a 2 +2ab+b 2 (3x+5) 2 = (3x) 2 +2(3x)(5)+(5) 2 = 9x x + 25 (3x-5) 2 =(3x+ - 5) 2 = (3x) 2 +2(3x)(-5)+(-5) 2 = 9x x + 25

 You may recall factoring numbers in the following way: So 60 written in factored form is 2·2·3·5 Polynomials can be factored in a similar fashion. Polynomials can be written in factored form as the product of linear factors.

(reverse of Distributive Property; factor out the common stuff) 6x – 9 = 2·3·x - 3·3 = 3(2x – 3) 5x 2 + 8x = 5·x·x + 2·2·2·x = x(5x+8) 10x 3 –15x 2 =2·5·x·x·x-3·5·x·x=5x 2 (2x-3) x 2 + 3x – 4 = x·x + 3·x - 2·2 = x 2 + 3x – 4 (nothing common)

 Group first two terms; make sure third term is addition; group last two terms  Common Monomial Factor both parentheses (inside stuff must be same in both parentheses)  Answer: (Outside stuff)·(Inside stuff)  5x 2 – 3x – 10x + 6 = (5x 2 – 3x) + ( – 10x + 6) = x(5x-3) – 2(5x – 3) = (x – 2)(5x – 3)

 List factor pairs of a; these are the possible coefficients of x in the two parentheses.  List factor pairs of c; these are the possible constant terms in the two parentheses.  Guess by combining the factor pairs of both a & c then compare the sum of the Outer and Inner multiplications to b.  If the check works you have your answer; if not guess again. a=2 b=7 c= - 15 Factor pairs of acac x 2 + 7x – 15 Guess #1: (x -1)(2x + 15) = 2x x – 2x – 15 = 2x x – 15 error Guess #2:(x + 5)(2x – 3) = 2x 2 -3x + 10x – 15 = 2x 2 + 7x – 15 correct Therefore (x + 5)(2x – 3) is your answer.

 Find two numbers, r & s, so that r + s = b and r · s = c  Answer: (x + r)(x + s) a=1 b=5 c= - 24 r+s = 5 r·s = · - 24= = · - 12= = · - 8= = · - 6= = · - 4= =2 8 · - 3= =5 12 · - 2= =10 24· - 1= =23 x 2 + 5x – 24 = (x+8)(x-3)

 Find square roots of both terms  Answer: (a + b)(a – b) 25x =(5x) 2 – (7) 2 =(5x + 7)(5x – 7) x2x2 x

Flowchart Common Monomial Difference of Two Squares Factor by Grouping unFOILing with shortcut Number of terms Does a = 1? yes no unFOILing without shortcut

Flowchart Common Monomial Difference of Two Squares Factor by Grouping unFOILing with shortcut Number of terms Does a = 1? yes no unFOILing without shortcut Perfect Square Trinomial Does 2ab part check? yes no Do you know square roots of first & last terms? yes no