Multiplying-and-dividing-polynomials.pptx

Multiplication and division

1. Multiplying and dividing polynomials

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. (3 )(-2x) = (3)(-2)(  x) = -6 2. (4 )(3 - 2x + 5) = (4 )(3 ) - (4 )(2x) + (4 )(5) Distributive Property = 12 - 8 + 20 Multiply the monomials  

4. Example: Multiplying Polynomials 3. (2x – 4)(7x + 5) = 2x(7x + 5) – 4(7x + 5) = 14 + 10x – 28x - 20 = 14 - 18x - 20  

5. Example: Multiplying Polynomials 4. = (3x + 4)(3x + 4) = (3x)(3x + 4) + 4(3x + 4) = 9 + 12x + 12x + 16 = 9 + 24x + 16  

6. Example: Multiplying Polynomials 5. (a + 2)( - 3 + 7) = a( - 3 + 7) + 2( - 3 + 7) = - 3 + 7a + 2 - 6 + 14 = - - 6 + 7a + 14  

7. Example: Multiplying Polynomials 6. = (5x – 2z)(5x – 2z) = (5x)(5x – 2z) – (2z)(5x – 2z) = 25 - 10xz – 10xz + 4 = 25 - 20xz + 4  

8. Example: Multiplying Polynomials 7. (2 + x – 1)( + 3x + 4) = (2 )( + 3x + 4) + (x)( + 3x + 4) – 1( + 3x + 4) = 2 + 6 + 8 + + 3 + 4x - - 3x – 4 = 2 + 7 + 10 + x - 4  

9. SPECIAL PRODUCTS When multiplying 2 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 (y – 12)(y + 4) ( y – 12)( y + 4) Product of first terms is . ( 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) = + 4y – 12y – 48 = - 8y - 48  

11. Example: Special Products 2. (2x – 4)(7x + 5) = 2x(7x) + 2x(5) – 4(7x) – 4(5) = + 10x – 28x – 20 = - 18x – 20  

12. SPECIAL PRODUCTS In the process of using the FOIL method on products of certain types of binomials, we see specific patterns that lead to special products. SQUARING A BINOMIAL MULTIPLYING THE SUM AND DIFFERENCE OF TWO TERMS (a + b)(a – b) = -  

13. Example: Special Products = (x + 2)(x + 2) = + 2x + 2x + 4 = + 4x + 4 = + 2 (x)(2) + = + 4x + 4  

14. Example: Special Products (x + 2)(x – 2) = + 2x – 2x – 4 = – 4 (a + b)(a – b) = - (x + 2)(x – 2) = - = - 4  

15. Example: Special Products = = (x + 4)(x – 4) =  

16. DIVIDING POLYNOMIALS When dividing a polynomial by a monomial, divide each term of the polynomial separately by the monomial. Example: = + - = -4 + 12 -  

17. DIVIDING POLYNOMIALS - Dividing a 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.

18. DIVIDING POLYNOMIALS One method of 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.

19. DIVIDING POLYNOMIALS Advantages and Disadvantages 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.

20. Using the long division: + 5x + 6  x – 1 =  

21. Using the synthetic division: First, take the polynomial, and write the coefficients ONLY inside in an upside–down division–type symbol. + 5x + 6  x – 1 =  

22. Using the synthetic division: Put the test zero, in our case x = 1, at the left, next to the (top) row of numbers: + 5x + 6  x – 1 =  

23. Using the synthetic division: 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: + 5x + 6  x – 1 =  

24. Using the synthetic division: Multiply this carry-down value by the test zero on the left, and carry the result up into the next column inside: + 5x + 6  x – 1 =  

25. Using the synthetic division: Add down the column: + 5x + 6  x – 1 =  

26. Using the synthetic division: Multiply the previous carry-down value by the test zero, and carry the new result up into the last column: + 5x + 6  x – 1 =  

27. Using the synthetic division: Add down the column: This last carry-down value is the remainder. + 5x + 6  x – 1 =  

28. You try! + 5x + 6  x – 1  

29. DIVIDING POLYNOMIALS DIVISION OF A MONOMIAL BY ANOTHER MONOMIAL = 2  2  x = 4x  

30. DIVIDING POLYNOMIALS DIVISION OF A POLYNOMIAL BY MONOMIAL 24x 3  – 12xy + 9x  3x = = 8 - 4y + 3  

31. DIVIDING POLYNOMIALS DIVISION OF A POLYNOMIAL BY BINOMIAL Divide:  3x 3  – 8x + 5 by x – 1

32. DIVIDING POLYNOMIALS DIVISION OF A POLYNOMIAL BY ANOTHER POLYNOMIAL Divide:   x 2  + 2x + 3x 3  + 5 by 1 + 2x + x 2