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Unit 6

Unit 6. Polynomials FOIL and Factoring. Terminology. Poly nomial – 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.

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Unit 6

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  1. Unit 6 Polynomials FOIL and Factoring

  2. Terminology • 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: nth degree

  3. Polynomial Addition and Subtraction • Addition: ignore parentheses and combine like terms. (2x3-5x2-7x+4) + (-6x3-2x2+x+6) = -4x3-7x2-6x+10 • Subtraction: distribute the minus to all terms in parentheses behind then combine like terms. • (2x3-5x2-7x+4) - (-6x3-2x2+x+6) • = (2x3-5x2-7x+4) + (--6x3--2x2-+x-+6) • = (2x3-5x2-7x+4) + (6x3+2x2-x-6) • = 8x3-3x2-8x-2

  4. Polynomial Multiplication • General Rule: Multiply every term of one polynomial by every term of the other • Special Polynomial Multiplications: • Distributive Property: (Monomial)·(any polynomial) • -3x2(5x2-6x+2) = (-3x2)(5x2)+ (-3x2)(-6x)+ (-3x2)(2) • = -15x4 + 18x3 - 6x2 • FOIL: (Binomial)·(binomial) • FirstOuterInner Last • (4x – 5)(2x + 7) = (4x)(2x) + (4x)(7) + (-5)(2x) + (-5)(7) • = 8x2+ 28x – 10x -35 • = 8x2 + 18x -35

  5. Punnett Squares method (3x + 2) (4x – 5) 3x + 2 4x -5

  6. 2 Special Patterns of FOIL Sum and Difference Pattern (a+b)(a–b) = a2–ab+ab–b2 = a2–b2 (3x+5)(3x-5) = (3x)2 – (5)2 = 9x2 - 25 • Square of a Binomial Pattern (a +b)2 = (a+b)(a+b) = a2+ab+ab+b2 = a2+2ab+b2 (3x+5)2 = (3x)2+2(3x)(5)+(5)2 = 9x2 + 30x + 25 (3x-5)2 =(3x+-5)2= (3x)2+2(3x)(-5)+(-5)2 = 9x2 - 30x + 25

  7. Factoring: Splitting polynomials into factors • You may recall factoring numbers in the following way: 60 6 10 2 3 2 5 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.

  8. Common Monomial Factoringalways check for first (reverse of Distributive Property; factor out the common stuff) 6x – 9 = 2·3·x - 3·3 = 3(2x – 3) 5x2 + 8x = 5·x·x + 2·2·2·x = x(5x+8) 10x3–15x2=2·5·x·x·x-3·5·x·x=5x2(2x-3) x2 + 3x – 4 = x·x + 3·x - 2·2 = x2 + 3x – 4 (nothing common)

  9. Factor by Grouping (4 terms) • 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) • 5x2 – 3x – 10x + 6 = (5x2 – 3x) + (–10x + 6) • = x(5x-3) – 2(5x – 3) • = (x – 2)(5x – 3)

  10. r & s method without shortcut(3 terms: ax2 + bx + c) • Find two numbers, r & s, so that r + s = b and r · s =a · c • Rewrite ax2 + bx + c as ax2 + rx+ sx+ c • Use factor by grouping rules to complete a=2 b=7 c=-15 r+s = 7 r·s = 2·-15 = -30 1·-30=-30 1+-30=-29 2 ·-15=-30 2+-15=-13 3 ·-10=-30 3+-10=-7 5 ·-6=-30 5+-6=-1 6 ·-5=-30 6+-5=1 10 ·-3=-30 10+-3=7 15 ·-2=-30 15+-2=13 30 ·-1=-30 30+-1=29 2x2 + 7x – 15 =2x2+ 10x – 3x -15 =(2x2 +10x) + (-3x – 15) =2x(x+5)-3(x+5) =(2x-3)(x+5)

  11. r & s method with shortcut(3 terms: x2 + bx + c;a=1) • Find two numbers, r & s, so that r + s = b and r · s=c • Answer: (x + r)(x + s) x2 + 5x – 24 = (x+8)(x-3) a=1 b=5 c=-24 r+s = 5 r·s = -24 1·-24=-24 1+-24=-23 2 ·-12=-24 2+-12=-10 3 ·-8=-24 3+-8=-5 4 ·-6=-24 4+-6=-2 6 ·-4=-24 6+-4=2 8 ·-3=-24 8+-3=5 12 ·-2=-24 12+-2=10 24·-1=-24 24+-1=23

  12. Difference of two squaresTwo terms: a2-b2 • Find square roots of both terms • Answer: (a + b)(a – b) 25x2 - 49 =(5x)2 – (7)2 =(5x + 7)(5x – 7)

  13. Perfect Square TrinomialsThree terms: a2±2ab+b2 • Find square roots of first and last terms • If 2ab matches the middle term, then answer is (a ± b)2; use sign of middle term. • If 2ab does not match then it is not a perfect square and you must use another method. 9x2-30x+25 =(3x)2–30x+(5)2 =(3x–5)2 b/c 2(3x)(5)=30x 4x2+36x+81 =(2x)2+36x+(9)2 =(2x+9)2 b/c 2(2x)(9)=36x

  14. Perfect Square TrinomialsThree terms: a2±2ab+b2 additional example x2+34x+64 =(x)2+34x+(8)2 ≠(x+8)2 b/c 2(x)(8)=16x ≠ 34x x2+34x+64 is not a perfect square; since a=1 use r&s with shortcut 32·2=64 and 32+2=34 so (x+32)(x+2)

  15. REVIEW: 6 factor types studied • Common Monomial: always look for first • Factor by Grouping: four terms • r&s without shortcut: three terms, a ≠1 • r&s with shortcut: three terms, a =1 • Difference of Two Squares: minus sign between two terms, know square roots of both • Perfect Square Trinomial: three terms, know square roots of first and last terms.

  16. Flowchart Common Monomial Difference of Two Squares Number of terms Factor by Grouping 2 3 4 r & s method with shortcut Does a = 1? yes no Do you know square roots of first and last terms? r & s method without shortcut Perfect Square Trinomial yes no Does 2ab part work? yes no

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