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Polynomials and Factoring

Polynomials and Factoring. What is a polynomial?. Any finite sum of terms. 6x² + 2x – 1 5x – 3 15. → trinomial of degree 2. → binomial of degree 1. → polynomial of degree 6. → monomial of degree 4. → monomial of degree 1. Adding & Subtracting.

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Polynomials and Factoring

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  1. Polynomials and Factoring

  2. What is a polynomial? • Any finite sum of terms. • 6x² + 2x – 1 • 5x – 3 • 15 → trinomial of degree 2 → binomial of degree 1 → polynomial of degree 6 → monomial of degree 4 → monomial of degree 1

  3. Adding & Subtracting • To add or subtract two polynomials, add or subtract the like terms. • Like terms Any two terms with the same variables and exponents

  4. Adding & Subtracting • Which of the following are like terms? 1) 7xy and 6x 2) 3y and -4y 3) 5x³ and 3x² 4) 4x²y³ and -2x³y² → not like terms (variables are different) → These are like terms → not like terms (exponents are different) → not like terms (exponents)

  5. Adding & Subtracting • Find the sum: (5x³ - x + 2x² + 7) + (3x² + 7 – 4x) + (4x² - 8 - x³) Starting with the highest degree, what are like terms? = 4x³ + 9x² - 5x + 6

  6. Adding & Subtracting • Find the sum: (2x² + x - 5) + (x + x² + 6) =3x² + 2x + 1

  7. Adding & Subtracting • Add the following polynomials: • (12x³ + 10) + (18x³ - 3x² + 6) • (4x² - 7x + 2) + (-x² + x – 2) • (8y³ + 4y² + 3y – 7) + (2y² - 6y + 4) • (x³ - 6x) + (2x³ + 9) + (4x² + x³)

  8. (12x³ + 10) + (18x³ - 3x² + 6) = 30x³ - 3x² + 16

  9. (4x² - 7x + 2) + (-x² + x – 2) = 3x² - 6x

  10. (8y³ + 4y² + 3y – 7) + (2y² - 6y + 4) = 8y³ - 3y + 6y² - 3

  11. (x³ - 6x) + (2x³ + 9) + (4x² + x³) = 4x³ - 6x + 4x² + 9

  12. Polynomials & Factoring

  13. Adding & Subtracting • Find the sum: (2x³ - 3x + 6x² + 7) + (-3x² - 7 + x) + (2x² - x³) Starting with the highest degree, what are like terms? = x³ + 5x² - 2x

  14. Subtracting • So far, the only operation we have used has been addition. • Most of the mistakes made on MCAS tests are on Subtraction problems

  15. Subtraction • Find the difference (5x² + 2x – 4) – (3x² - 3x + 6) You must distribute the negative sign over the polynomial in the immediately parentheses to the right 5x² + 2x – 4 -3x² -6 = 2x² + 8x - 10 + 6x

  16. Subtracting • Find the difference between the polynomials. (5h² + 4h + 8) – (3h² + h + 3) → Re-write the equation after distributing the minus sign 5h² + 4h + 8 + -3h² - h -3 = 2h² + 3h + 5

  17. Subtracting (-7z³ + 2z – 7) – (2z³ - z – 3) -7z³ + 2z - 7 + -2z³ + z + 3 = -9z³ + 3z - 4

  18. Perimeter • The perimeter of a polygon is the sum of the lengths of its sides. Write a simplified expression for the perimeter of this polygon. x + 2 4x – 2 2x + 3 3x + 15 Per. = 2x + 3 + x + 2 + 4x – 2 + 3x + 15 Per. = 10x + 18

  19. Perimeter • The perimeter of a polygon is the sum of the lengths of its sides. Write a simplified expression for the perimeter of this polygon. x + 4 x - 3 x – 3 x - 1 Per. = x - 3 + x + 4 + x – 3 + x - 1 Per. = 4x - 3

  20. Polynomials & Factoring

  21. So far we have: • Added polynomials (4x³ + 2x² + 3) + (2x³ - x² - 2) • Subtracted polynomials (3x² + 4x + 8) – (2x² + 3x – 5) 3x² + 4x + 8 6x³ + x² + 1 - 2x² - 3x + 5 + x + 13 x²

  22. Multiplying Polynomials • When we multiply two binomials (x + 2) (2x – 4) Use the F.O.I.L. method F - First O - Outer I - Inner L - Last

  23. F.O.I.L (x + 2) (2x – 4) = 2x² - 4x + 4x - 8 = 2x² - 8 F - First x • 2x O - Outer x • - 4 I - Inner 2 • 2x L - Last 2 • -4

  24. F.O.I.L (x + 2) (x – 3) = x² + -3x + 2x + -6 = x² + -1x - 6 x • x = x² x • -3 = -3x 2 • x = 2x 2 • -3 = -6

  25. Multiply the following binomials. • (2x – 3) (x + 4) • (3x + 1) (x + 5) • (x – 4) (x + 7) • (4x + 4) (2x – 3)

  26. F.O.I.L (2x - 3) (x + 4) = 2x² + 8x + -3x + -12 = 2x² + 5x - 12 2x • x = 2x² 2x • 4 = 8x -3 • x = -3x -3 • 4 = -12

  27. F.O.I.L (3x + 1) (x + 5) = 3x² + 15x + x + 5 = 3x² + 16x + 5 3x • x = 3x² 3x • 5 = 15x 1 • x = x 1 • 5 = 5

  28. F.O.I.L (x - 4) (x + 7) = x² + 7x + -4x - 28 = x² + 3x - 28 x • x = x² x • 7 = 7x -4 • x = -4x -4 • 7 = -28

  29. F.O.I.L (4x + 4) (2x - 3) = 8x² + -12x + 8x - 12 = 8x² - 4x - 12 4x • 2x = 8x² 4x • -3 = -12x 4 • 2x = 8x 4 • -3 = -12

  30. Polynomials & Factoring

  31. This week: • Added polynomials • Subtracted polynomials • Multiplied binomials (F.O.I.L.)

  32. Adding Polynomials (x² + 3x – 4) + (3x² + 5x – 8) = 4x² + 8x - 12 (2x² – 7) + (4x² + 3x + 3) = 6x² + 3x - 4

  33. Subtracting Polynomials (5x² + 2x – 7) -(2x² + 4x – 3) = 5x² + 2x - 7 - 2x² -4x + 3 = 3x² - 2x - 4

  34. F.O.I.L (2x + 3) (4x - 1) = 8x² + -2x + 12x - 3 = 8x² + 10x - 3 2x • 4x = 8x² 2x • -1 = -2x 3• 4x = 12x 3• -1 = -3

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