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Review 10.1-10.4 Polynomials

Review 10.1-10.4 Polynomials. Vocabulary. Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x 2 yw 3 , -3, a 2 b 3 , and 3yz are all monomials. Polynomials – one or more monomials added or subtracted

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Review 10.1-10.4 Polynomials

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  1. Review 10.1-10.4Polynomials

  2. Vocabulary • Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials. • Polynomials – one or more monomials added or subtracted • 4x + 6x2, 20xy - 4, and 3a2 - 5a + 4 are all polynomials.

  3. Like Terms Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers. Which terms are like? 3a2b, 4ab2, 3ab, -5ab2 4ab2 and -5ab2 are like. Even though the others have the same variables, the exponents are not the same. 3a2b = 3aab, which is different from 4ab2 = 4abb.

  4. Like Terms Constants are like terms. Which terms are like? 2x, -3, 5b, 0 -3 and 0 are like. Which terms are like? 3x, 2x2, 4, x 3x and x are like. Which terms are like? 2wx, w, 3x, 4xw 2wx and 4xw are like.

  5. Classifying Polynomials A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Identify the following polynomials: Classified by degree Classified by number of terms Polynomial Degree constant monomial 6 0 linear monomial –2x 1 linear binomial 1 3x + 1 quadratic trinomial –x2 + 2x – 5 2 cubic binomial 3 4x3 – 8x quartic polynomial 4 2x4 – 7x3 – 5x + 1

  6. Adding Polynomials Add: (x2 + 3x + 1) + (4x2 +5) Step 1: Underline like terms: (x2 + 3x + 1) + (4x2 +5) Notice: ‘3x’ doesn’t have a like term. Step 2: Add the coefficients of like terms, do not change the powers of the variables: (x2 + 4x2) + 3x + (1 + 5) 5x2 + 3x + 6

  7. Adding Polynomials Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms! (x2 + 3x + 1) + (4x2 +5) (x2 + 3x + 1) + (4x2 +5) 5x2 + 3x + 6 Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2) (2a2 + 3ab + 4b2) + (7a2 + ab + -2b2) (2a2+3ab+4b2) + (7a2+ab+-2b2) 9a2 + 4ab + 2b2

  8. Adding Polynomials • Add the following polynomials; you may stack them if you prefer:

  9. Subtracting Polynomials Subtract: (3x2 + 2x + 7) - (x2 + x + 4) Step 1: Change subtraction to addition (Keep-Change-Change.). (3x2 + 2x + 7) + (- x2 + - x + - 4) Step 2: Underline OR line up the like terms and add. (3x2 + 2x + 7) + (- x2 + - x + - 4) 2x2 + x + 3

  10. Subtracting Polynomials • Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:

  11. 1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a) Group your like terms. 9y - 3y - 7x + 8x + 15a - 8a 6y + x + 7a

  12. 2. Add the following polynomials:(3a2 + 3ab - b2) + (4ab + 6b2) Combine your like terms. 3a2 + 3ab + 4ab - b2 + 6b2 3a2 + 7ab + 5b2

  13. Add the polynomials.+ Y X X2 Y X XY Y X Y 1 1 Y Y 1 1 1 1 1 1 Y • x2 + 3x + 7y + xy + 8 • x2 + 4y + 2x + 3 • 3x + 7y + 8 • x2 + 11xy + 8

  14. 3. Add the following polynomials using column form:(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2) Line up your like terms. 4x2 - 2xy + 3y2 + -3x2 - xy + 2y2 _________________________ x2 - 3xy + 5y2

  15. 4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a) Rewrite subtraction as adding the opposite. (9y - 7x + 15a) + (+ 3y - 8x + 8a) Group the like terms. 9y + 3y - 7x - 8x + 15a + 8a 12y - 15x + 23a

  16. 5. Subtract the following polynomials:(7a - 10b) - (3a + 4b) Rewrite subtraction as adding the opposite. (7a - 10b) + (- 3a - 4b) Group the like terms. 7a - 3a - 10b - 4b 4a - 14b

  17. 6. Subtract the following polynomials using column form:(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2) Line up your like terms and add the opposite. 4x2 - 2xy + 3y2 + (+ 3x2+ xy - 2y2) -------------------------------------- 7x2 - xy + y2

  18. Find the sum or difference.(5a – 3b) + (2a + 6b) • 3a – 9b • 3a + 3b • 7a + 3b • 7a – 3b

  19. Find the sum or difference.(5a – 3b) – (2a + 6b) • 3a – 9b • 3a + 3b • 7a + 3b • 7a – 9b

  20. Adding Polynomials + Find the sum. Write the answer in standard format. (5x3 – x + 2x2 + 7) + (3x2 + 7 – 4x) + (4x2 – 8 – x3) SOLUTION Vertical format: Write each expression in standard form. Align like terms. 5x3 + 2x2 – x + 7 3x2 – 4x + 7 – x3+ 4x2 – 8 4x3 + 9x2 – 5x + 6

  21. Adding Polynomials Find the sum. Write the answer in standard format. (2x2 + x – 5) + (x + x2 + 6) SOLUTION Horizontal format: Add like terms. (2x2 + x – 5) + (x + x2 + 6) = (2x2 +x2) + (x + x) + (–5 + 6) = 3x2 + 2x+ 1

  22. Subtracting Polynomials + – Find the difference. (–2x3 + 5x2 – x+ 8) – (–2x2 + 3x – 4) SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2x3 + 5x2 – x + 8 No change –2x3 + 5x2 – x + 8 –2x3 + 3x – 4 2x3– 3x+ 4 Add the opposite

  23. Subtracting Polynomials + Find the difference. (–2x3 + 5x2 – x+ 8) – (–2x2 + 3x – 4) SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2x3 + 5x2 – x + 8 –2x3 + 5x2 – x + 8 – –2x3 + 3x – 4 2x3– 3x+ 4 5x2 – 4x + 12

  24. Subtracting Polynomials Find the difference. (3x2 – 5x + 3) – (2x2 – x – 4) SOLUTION Use a horizontal format. (3x2 – 5x + 3) – (2x2 – x – 4) = (3x2 – 5x + 3) + (–1)(2x2 – x – 4) = (3x2 – 5x + 3) – 2x2 + x + 4 = (3x2– 2x2) + (– 5x +x) + (3+ 4) = x2 – 4x+ 7

  25. MultiplyingPolynomials Distribute and FOIL

  26. Polynomials * Polynomials Multiplying a Polynomial by another Polynomial requires more than one distributing step. Multiply: (2a + 7b)(3a + 5b) Distribute 2a(3a + 5b) and distribute 7b(3a + 5b): 6a2 + 10ab 21ab + 35b2 Then add those products, adding like terms: = 6a2 + 31ab + 35b2 6a2 + 10ab + 21ab + 35b2

  27. Polynomials * Polynomials An alternative is to stack the polynomials and do long multiplication. (2a + 7b) x (3a + 5b) (2a + 7b)(3a + 5b) (2a + 7b) x (3a + 5b) Multiply by 5b, then by 3a: When multiplying by 3a, line up the first term under 3a. 21ab + 35b2 + 6a2 + 10ab Add like terms: 6a2 + 31ab + 35b2

  28. Polynomials * Polynomials Multiply the following polynomials:

  29. (x + 5) x (2x + -1) + (3w + -2) x (2w + -5) + Polynomials * Polynomials -x + -5 2x2 + 10x 2x2 + 9x + -5 -15w + 10 6w2 + -4w 6w2 + -19w + 10

  30. (2a2 + a + -1) x (2a2 + 1) + Polynomials * Polynomials 2a2 + a + -1 4a4 + 2a3 + -2a2 4a4 + 2a3 + a + -1

  31. Types of Polynomials • We have names to classify polynomials based on how many terms they have: Monomial: a polynomial with one term Binomial: a polynomial with two terms Trinomial: a polynomial with three terms

  32. (2x + -3)(4x + 5) = 8x2 + 10x + -12x + -15 = 8x2 + -2x + -15 F.O.I.L. There is an acronym to help us remember how to multiply two binomials without stacking them. (2x + -3)(4x + 5) F : Multiply the First term in each binomial. 2x • 4x = 8x2 O : Multiply the Outer terms in the binomials. 2x • 5 = 10x I : Multiply the Inner terms in the binomials. -3 • 4x = -12x L : Multiply the Last term in each binomial. -3 • 5 = -15

  33. F.O.I.L. Use the FOIL method to multiply these binomials: 1) (3a + 4)(2a + 1) 2) (x + 4)(x - 5) 3) (x + 5)(x - 5) 4) (c - 3)(2c - 5) 5) (2w + 3)(2w - 3)

  34. F.O.I.L. Use the FOIL method to multiply these binomials: 1) (3a + 4)(2a + 1) = 6a2 + 3a + 8a + 4 = 6a2 + 11a + 4 2) (x + 4)(x - 5) = x2 + -5x + 4x + -20 = x2 + -1x + -20 3) (x + 5)(x - 5) = x2 + -5x + 5x + -25 = x2 + -25 4) (c - 3)(2c - 5) = 2c2 + -5c + -6c + 15 = 2c2 + -11c + 15 5) (2w + 3)(2w - 3) = 4w2 + -6w + 6w + -9 = 4w2 + -9

  35. There are three techniques you can use for multiplying polynomials. The best part about it is that they are all the same! Huh? Whaddaya mean? It’s all about how you write it…Here they are! • Distributive Property • FOIL • Box Method Sit back, relax (but make sure to write this down), and I’ll show ya!

  36. 1) Multiply. (2x + 3)(5x + 8) Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8). 10x2 + 16x + 15x + 24 Combine like terms. 10x2 + 31x + 24 A shortcut of the distributive property is called the FOIL method.

  37. The FOIL method is ONLY used when you multiply 2 binomials. It is an acronym and tells you which terms to multiply. 2) Use the FOIL method to multiply the following binomials:(y + 3)(y + 7).

  38. (y + 3)(y + 7). F tells you to multiply the FIRST terms of each binomial. y2

  39. (y + 3)(y + 7). O tells you to multiply the OUTER terms of each binomial. y2+7y

  40. (y + 3)(y + 7). I tells you to multiply the INNER terms of each binomial. y2 + 7y +3y

  41. (y + 3)(y + 7). L tells you to multiply the LAST terms of each binomial. y2 + 7y + 3y + 21 Combine like terms. y2 + 10y + 21

  42. Remember, FOIL reminds you to multiply the: First terms Outer terms Inner terms Last terms

  43. The third method is the Box Method. This method works for every problem! Here’s how you do it. Multiply (3x – 5)(5x + 2) Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where. This will be modeled in the next problem along with FOIL.

  44. 3) Multiply (3x - 5)(5x + 2) 15x2 First terms: Outer terms: Inner terms: Last terms: Combine like terms. 15x2 - 19x – 10 +6x -25x 15x2 -25x -10 +6x -10 You have 3 techniques. Pick the one you like the best!

  45. First terms: Outer terms: Inner terms: Last terms: Combine like terms. 21p2 – 34p + 8 4) Multiply (7p - 2)(3p - 4) 21p2 -28p -6p 21p2 -6p +8 -28p +8

  46. Multiply (y + 4)(y – 3) • y2 + y – 12 • y2 – y – 12 • y2 + 7y – 12 • y2 – 7y – 12 • y2 + y + 12 • y2 – y + 12 • y2 + 7y + 12 • y2 – 7y + 12

  47. Multiply (2a – 3b)(2a + 4b) • 4a2 + 14ab – 12b2 • 4a2 – 14ab – 12b2 • 4a2 + 8ab – 6ba – 12b2 • 4a2 + 2ab – 12b2 • 4a2 – 2ab – 12b2

  48. 5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property. 2x(x2 - 5x + 4) - 5(x2 - 5x + 4) 2x3 - 10x2 + 8x - 5x2 + 25x - 20 Group and combine like terms. 2x3 - 10x2 - 5x2 + 8x + 25x - 20 2x3 - 15x2 + 33x - 20

  49. 5) Multiply (2x - 5)(x2 - 5x + 4)You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. 2x3 -10x2 +8x Almost done! Go to the next slide! -5x2 +25x -20

  50. 5) Multiply (2x - 5)(x2 - 5x + 4)Combine like terms! 2x3 -10x2 +8x -5x2 +25x -20 2x3 – 15x2 + 33x - 20

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