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Chapter 8 Polynomials and Factoring

Chapter 8 Polynomials and Factoring. 8.1 Add and Subtract Polynomials. Monomial. A number, variable, or product of the two. EX: 4, y, or 4y Degree of a monomial – the sum of all of the exponents of the variables EX: 4x 2 y. Polynomial. A sum of monomials EX: 2x 2 + 5x + 7

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Chapter 8 Polynomials and Factoring

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  1. Chapter 8Polynomials and Factoring

  2. 8.1Add and Subtract Polynomials

  3. Monomial • A number, variable, or product of the two. • EX: 4, y, or 4y • Degree of a monomial – the sum of all of the exponents of the variables • EX: 4x2y

  4. Polynomial • A sum of monomials • EX: 2x2 + 5x + 7 • Degree of a polynomial – the greatest degree of its terms • EX: 2x2 + 5x + 7 • Leading coefficient – the coefficient of the first term when the polynomial is written from greatest exponent to least (descending order) • EX: 2x2 + 5x + 7

  5. EX: • Write the polynomial so that the exponents decrease from left to right. Identify the degree and the leading coefficient. • 7 – 5y3 • -5 + 2x2 + x3 – 7x

  6. Binomial – a polynomial with 2 terms • EX: 4x + 9 • Trinomial – a polynomial with 3 terms • EX: x2 + 7x – 9

  7. To add polynomials - • Add like terms • REMEMBER: You can only add if the variable AND the exponent are the same. • EX: Find the sum. • (6a2 – 4) + (2a2 – 9) • (5x3 + 4x – 2x) + (4x2 + 3x3 – 6)

  8. To subtract polynomials - • Add the opposite of every term in the polynomial being subtracted • Switch all signs • EX: Find the difference. • (4n2 + 5) – (-2n2 + 2n – 4) • (4x2 – 7x) – (5x2 + 4x – 9)

  9. EX: • Write a polynomial that represents the perimeter of the figure. • All sides added up.

  10. EX: • Major League Baseball teams are divided into two leagues. During the period 1995-2001, the attendance (in thousands) at National (N) and American (A) games can be modeled by: • N = -488t2 + 5430t + 24,700 • A = -318t2 + 3040t + 25,600 • Where t is the number of years since 1995. About how many total people attended games in 2001?

  11. 8.2 Multiplying Polynomials

  12. To multiply polynomials: • Distribute everything in the first polynomial to everything in the second. • REMEMBER: When you multiply like bases, add the exponents.

  13. EX: Find the product. • x(2x3 - 7x2 + 4) • (x – 2)(x2 +2x + 1) • (3y2 – y + 5)(2y – 3)

  14. FOIL Method • When multiplying two binomials, you can use the FOIL Method. • Multiply: • First • Outer • Inner • Last

  15. EX: Find the product. • (4b – 5)(b – 2) • (6n – 1)(n + 5) • (2x + 3y)2

  16. EX: Simplify the expression. • -3x2(x+ 11) – (4x – 5)(3x – 2)

  17. EX: • Write a polynomial that represents the area of the shaded region.

  18. EX: • You are planning to build a walkway that surrounds a rectangular garden, as shown. The width of the walkway is the same on every side. • Write a polynomial that represents the combined area of the garden and the walkway. • Find the combined area when the width of the walkway is 4 feet. 10 ft 9 ft x

  19. 8.4Solve Polynomial Equations in Factored Form

  20. Factoring • To factor a polynomial – write it as a product • To Factor a Polynomial: • Step 1: Look for a common monomial • A monomial that can be divided evenly out of each term in the polynomial • Take out the common monomial and multiply it by what is left over

  21. EX: Factor out the greatest common monomial factor. • 12x + 42y • 4x4 + 24x3 • 15n3 – 25n • 8a2b – 6ab2 + 4ab

  22. Zero-Product Property • If ab = 0, then a = 0 or b = 0. • This property is used to solve an equation when one side is zero and the other side is a product. • EX: (x-3)(2x + 7) = 0 • Solutions are called roots.

  23. To solve an equation by factoring: • 1) Put the equation in standard form set equal to zero. • 2) Factor • 3) Set each factor equal to zero and solve.

  24. EX: Solve the equation. • (2y + 5)(7y – 5)= 0 • a2 + 5a = 0 • -28m2 = 8m

  25. Vertical Motion Model • Models the height of a projectile • An object thrown in the air with only the force due to gravity acting on it. • http://phet.colorado.edu/sims/projectile-motion/projectile-motion_en.html • h = -16t2 + vt + s • h = height of the object (in feet) • t = time the object is in the air (in seconds) • v = initial vertical velocity (in feet per second) • s = initial height (in feet)

  26. EX: • A dolphin jumped out of the water with an initial vertical velocity of 32 feet per second. After how many seconds did the dolphin enter the water? • After 1 second, how high was the dolphin above the water?

  27. 8.5Factor x2 + bx + c

  28. Review: • Multiplying two binomials results in a trinomial. • EX: (3x + 2)(x – 4) • Therefore, we will factor a trinomial into two binomials.

  29. To factor a polynomial: • Step 1: Look for a common monomial. • Step 2: If you have a trinomial, factor it into two binomials. • The 1st terms in each binomial must multiply to get the 1st term in the trinomial. • The 2nd term in each binomial must add to get the 2nd term’s coefficient in the trinomial and must multiply to get the 3rd term in the trinomial. • NOTE: Pay attention to signs.

  30. EX: Factor the trinomial. • x2 + 3x + 2 • t2 + 9t + 14 • x2 – 4x + 3 • t2 – 8t + 12 • m2 + m – 20 • w2 + 6w – 16 • x2 – 4xy + 4y2 • m2 – mn – 42n2

  31. EX: Solve the equation (by factoring). • x2 – 2x = 24 • x2 – 2x – 8 = 7 • s(s + 1) = 72

  32. EX: Find the dimensions of the rectangle. • 8.5 Ex #43

  33. EX: • You are designing a flag for the SMCC football team with the dimensions shown. The flag requires 80 square feet of fabric. Find the width w of the flag. SMCC

  34. 8.6Factor ax2 + bx + c

  35. Steps to Factoring: • 1) Look for a common monomial • If the leading coefficient is negative, factor out a negative one. • 2) If you have a trinomial, factor it into two binomials.

  36. Test to factor a trinomial when the leading coefficient is not 1: • In your two binomials: • The 1st terms must multiply to get the 1st term in the trinomial. • The 2nd terms must multiply to get the 3rd term in the trinomial. • Outer times outer plus inner times inner must equal the middle coefficient in the trinomial.

  37. EX: Factor the trinomial. • 3x2 + 8x + 4 • 4x2 – 9x + 5 • 2x2 + 13x – 7 • -2x2 – 5x – 3 • -5m2 + 6m – 1 • -4n2 – 16n – 15

  38. EX: Solve the equation (by factoring). • 8x2 – 2x = 3 • b(20b – 3) = 2 • 6x2 – 15x = 99

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