Downloaded 61 times
More Related Content
Similar to Multiplying-and-dividing-polynomials.pptx
Multiplying-and-dividing-polynomials.pptx
- 1.
- 2. MULTIPLICATION OF POLYNOMIALS •If all the polynomials are monomials, use the associative and commutative property. • If any of the polynomials are not monomials, use the distributive property before the associative and commutative properties. Then combine like terms.
- 3. Example: Multiplying Polynomials 1.(3𝑥2)(-2x) = (3)(-2)(𝑥2 x) = -6 𝑥3 2. (4𝑥2)(3𝑥2 - 2x + 5) = (4𝑥2 )(3𝑥2 ) - (4𝑥2 )(2x) + (4𝑥2 )(5) Distributive Property = 12𝑥4 - 8𝑥3 + 20𝑥2 Multiply the monomials
- 4. Example: Multiplying Polynomials 3.(2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5) = 14 𝑥2 + 10x – 28x - 20 = 14 𝑥2 - 18x - 20
- 5. Example: Multiplying Polynomials 4.3𝑥 + 4 2 = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4) = 9 𝑥2 + 12x + 12x + 16 = 9 𝑥2 + 24x + 16
- 6. Example: Multiplying Polynomials 5.(a + 2)(𝑎3 - 3 𝑎2 + 7) = a(𝑎3 - 3 𝑎2 + 7) + 2(𝑎3 - 3 𝑎2 + 7) = 𝑎4 - 3 𝑎3 + 7a + 2 𝑎3 - 6 𝑎2 + 14 = 𝑎4 - 𝑎3 - 6 𝑎2 + 7a + 14
- 7. Example: Multiplying Polynomials 6.5𝑥 − 2𝑧 2 = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – (2z)(5x – 2z) = 25 𝑥2 - 10xz – 10xz + 4 𝑧2 = 25 𝑥2 - 20xz + 4 𝑧2
- 8. Example: Multiplying Polynomials 7.(2 𝑥2 + x – 1)(𝑥2 + 3x + 4) = (2 𝑥2)(𝑥2 + 3x + 4) + (x)(𝑥2 + 3x + 4) – 1(𝑥2 + 3x + 4) = 2 𝑥4 + 6 𝑥3 + 8 𝑥2 + 𝑥3 + 3 𝑥2 + 4x - 𝑥2 - 3x – 4 = 2 𝑥4 + 7 𝑥3 + 10 𝑥2 + x - 4
- 9. SPECIAL PRODUCTS When multiplying2 binomials, the distributive property can be easily remembered as the FOIL method. F – product of FIRST term O – product of OUTSIDE term I – product of INSIDE term L – product of LAST term
- 10. Example: Special Products 1.(y – 12)(y + 4) (y – 12)(y + 4) Product of first terms is 𝑦2 . (y – 12)(y + 4) Product of outside terms is 4y. (y – 12)(y + 4) Product of inside terms is -12y. (y – 12)(y + 4) Product of last terms is -48. (y – 12)(y + 4) = 𝑦2 + 4y – 12y – 48 = 𝑦2 - 8y - 48
- 11. Example: Special Products 2.(2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5) = 14𝑥2 + 10x – 28x – 20 = 14𝑥2 - 18x – 20
- 12. SPECIAL PRODUCTS In theprocess of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products. SQUARING A BINOMIAL 𝒂 + 𝒃 𝟐 = 𝒂𝟐 + 𝟐𝒂𝒃 + 𝒃𝟐 𝒂 − 𝒃 𝟐 = 𝒂𝟐 − 𝟐𝒂𝒃 + 𝒃𝟐 MULTIPLYING THE SUM AND DIFFERENCE OF TWO TERMS (a + b)(a – b) = 𝒂𝟐 - 𝒃𝟐
- 13. Example: Special Products 1.𝑥 + 2 2 = (x + 2)(x + 2) = 𝑥2 + 2x + 2x + 4 = 𝑥2 + 4x + 4 𝒂 + 𝒃 𝟐 = 𝒂𝟐 + 𝟐𝒂𝒃 + 𝒃𝟐 𝑥 + 2 2 = (𝑥)2 + 2 (x)(2) + (2)2 = 𝑥2 + 4x + 4
- 14. Example: Special Products 1.(x + 2)(x – 2) = 𝑥2 + 2x – 2x – 4 = 𝑥2 – 4 (a + b)(a – b) = 𝒂𝟐 - 𝒃𝟐 (x + 2)(x – 2) = (𝑥)2 - (2)2 = 𝒙𝟐 - 4
- 15. Example: Special Products 1.𝑥 − 2 2 = 2. 𝑥 + 3 2 = 3. (x + 4)(x – 4) =
- 16. DIVIDING POLYNOMIALS When dividinga polynomial by a monomial, divide each term of the polynomial separately by the monomial. Example: −12𝑎3+36𝑎 −15 3𝑎 = −12𝑎3 3𝑎 + 36𝑎 3𝑎 - 15 3𝑎 = -4𝑎2 + 12 - 5 𝑎
- 17. DIVIDING POLYNOMIALS - Dividinga polynomial by a polynomial other than a monomial uses a “long division” technique that is like the process known as long division in dividing two numbers.
- 21. DIVIDING POLYNOMIALS One methodof dividing polynomials is the SYNTHETIC DIVISION. Synthetic division is generally used, however, not for dividing out factors but for finding zeroes (or roots) of polynomials.
- 22. DIVIDING POLYNOMIALS Advantages andDisadvantages of Synthetic Division Method The advantages of using the synthetic division method are: • It requires only a few calculation steps • The calculation can be performed without variables • Unlike the polynomial long division method, this method is a less error-prone method The only disadvantage of the synthetic division method is that this method is only applicable if the divisor of the polynomial expression is a linear factor.
- 23. Using the longdivision: 𝒙𝟐+ 5x + 6 x – 1 =
- 24. Using the syntheticdivision: First, take the polynomial, and write the coefficients ONLY inside in an upside– down division–type symbol. 𝒙𝟐+ 5x + 6 x – 1 =
- 25. Using the syntheticdivision: Put the test zero, in our case x = 1, at the left, next to the (top) row of numbers: 𝒙𝟐+ 5x + 6 x – 1 =
- 26. Using the syntheticdivision: Take the first number that's on the inside, the number that represents the polynomial's leading coefficient, and carry it down, unchanged, to below the division symbol: 𝒙𝟐+ 5x + 6 x – 1 =
- 27. Using the syntheticdivision: Multiply this carry-down value by the test zero on the left, and carry the result up into the next column inside: 𝒙𝟐+ 5x + 6 x – 1 =
- 28. Using the syntheticdivision: Add down the column: 𝒙𝟐+ 5x + 6 x – 1 =
- 29. Using the syntheticdivision: Multiply the previous carry-down value by the test zero, and carry the new result up into the last column: 𝒙𝟐+ 5x + 6 x – 1 =
- 30. Using the syntheticdivision: Add down the column: This last carry-down value is the remainder. 𝒙𝟐+ 5x + 6 x – 1 =
- 31. You try! 1. 𝑥2 +5x + 6 x – 1 2. 2𝑥3 −5𝑥2+3𝑥+7 𝑥 −2 3. 2𝑥3+ 5𝑥2+9 𝑥+3
- 32. DIVIDING POLYNOMIALS DIVISION OFA MONOMIAL BY ANOTHER MONOMIAL 40𝑥2 10𝑥 = 2 ∙ 2 ∙ 2 ∙ 5 ∙ 𝑥 ∙ 𝑥 2 ∙ 5 ∙ 𝑥 = 2 2 x = 4x
- 33. DIVIDING POLYNOMIALS DIVISION OFA POLYNOMIAL BY MONOMIAL 24x3 – 12xy + 9x 3x = 24x3 3𝑥 − 12𝑥𝑦 3𝑥 + 9𝑥 3𝑥 = 8𝑥2 - 4y + 3
- 34. DIVIDING POLYNOMIALS DIVISION OFA POLYNOMIAL BY BINOMIAL Divide: 3x3 – 8x + 5 by x – 1
- 35. DIVIDING POLYNOMIALS DIVISION OFA POLYNOMIAL BY ANOTHER POLYNOMIAL Divide: x2 + 2x + 3x3 + 5 by 1 + 2x + x2