Multiplying Polynomials

1. Multiplying Polynomials Lesson 5-1

2. Exponential Form • When a number, variable, or expression is raised to a power, the number, variable, or expression is called the base and the power is called the exponent.

3. What is an Exponent? • An exponent means that you multiply the base by itself that many times. • For example x4 = x ● x ● x ● x 26 = 2 ● 2 ● 2 ● 2 ● 2 ● 2 = 64

4. The Invisible Exponent • When an expression does not have a visible exponent its exponent is understood to be 1.

5. Summary of Exponent Rules w/ examples • When multiplying two expressions with the same base you add their exponents. • For example

6. Exponent Rule #1 • Try it on your own:

7. Exponent Rule #2 • When dividing two expressions with the same base you subtract their exponents. • For example

8. Exponent Rule #2 • Try it on your own:

9. Exponent Rule #3 • When raising a power to a power you multiply the exponents • For example

10. Exponent Rule #3 • Try it on your own

11. Note • When using this rule the exponent can not be brought in the parenthesis if there is addition or subtraction You would have to use FOIL in these cases

12. Exponent Rule #4 • When a product is raised to a power, each piece is raised to the power • For example

13. Exponent Rule #4 • Try it on your own

14. Note • This rule is for products only. When using this rule the exponent can not be brought in the parenthesis if there is addition or subtraction You would have to use FOIL in these cases

15. Exponent Rule #5 • When a quotient is raised to a power, both the numerator and denominator are raised to the power • For example

16. Exponent Rule #5 • Try it on your own

17. Zero Exponent • When anything, except 0, is raised to the zero power it is 1. • For example ( if a ≠ 0) ( if x ≠ 0)

18. Zero Exponent • Try it on your own ( if a ≠ 0) ( if h ≠ 0)

19. Negative Exponents • If b ≠ 0, then • For example

20. Negative Exponents • If b ≠ 0, then • Try it on your own:

21. Negative Exponents • The negative exponent basically flips the part with the negative exponent to the other half of the fraction.

22. Math Manners • Simplifying Monomials: For a monomial to be completely simplified • No powers of powers • Each variable appears only once • All fractions are simplified • there should not be any negative exponents

23. Examples Pg. 303

24. Polynomial Vocabulary Term – a number or a product of a number and variables raised to powers Coefficient – numerical factor of a term Constant – term which is only a number Polynomial is a sum of terms involving variables raised to a whole number exponent, with no variables appearing in any denominator.

25. Types of Polynomials Monomial is a polynomial with 1 term. Binomial is a polynomial with 2 terms. Trinomial is a polynomial with 3 terms.

26. Degrees Degree of a term To find the degree, take the sum of the exponents on the variables contained in the term. The degree of a constant is 0. The degree of the term 5a4b3c is 8 (remember that c can be written as c1). Degree of a polynomial To find the degree, take the largest degree of any term of the polynomial. Degree of 9x3 – 4x2 + 7 is 3.

27. = x2y + 10x2y + xy + xy – y – 2y (Like terms are grouped together) Combining Like Terms Like terms are terms that contain exactly the same variables raised to exactly the same powers. Warning! Only like terms can be combined through addition and subtraction. Example • Combine like terms to simplify. x2y + xy – y + 10x2y – 2y + xy = (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y= 11x2y + 2xy – 3y

28. Adding and Subtracting Polynomials Adding Polynomials Combine all the like terms. Subtracting Polynomials Change the signs of the terms of the polynomial being subtracted, and then combine all the like terms.

29. Adding and Subtracting Polynomials Example = 3a2 – 6a + 11 Add or subtract each of the following, as indicated. 1) (3x – 8) + (4x2 – 3x +3) = 4x2 + 3x – 3x – 8 + 3 = 3x – 8 + 4x2 – 3x + 3 = 4x2 – 5 2) 4 – (– y – 4) = 4 + y + 4 = y + 8 = y + 4 + 4 3) (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7) = – a2 + 1 – a2 + 3 + 5a2 – 6a + 7 = – a2 – a2 + 5a2 – 6a + 1 + 3 + 7

30. Multiplying Polynomials Multiplying polynomials • If all of the polynomials are monomials, use the associative and commutative properties. • If any of the polynomials are not monomials, use the DISTRIBUTIVE PROPERTY, then COMBINE LIKE TERMS.

31. Examples Multiplying Polynomials (pg. 306)

32. Dividing Polynomials Dividing a polynomial by a monomial Divide each term of the polynomial separately by the monomial. Example

33. Dividing Polynomials Dividing a polynomial by a polynomial other than a monomial uses a “long division” technique that is similar to the process known as long division in dividing two numbers, which is reviewed on the next slide.

34. We then write our result as Dividing Polynomials Divide 43 into 72. Multiply 1 times 43. Subtract 43 from 72. Bring down 5. Divide 43 into 295. Multiply 6 times 43. Subtract 258 from 295. Bring down 6. Divide 43 into 376. Multiply 8 times 43. Subtract 344 from 376. Nothing to bring down.

35. - 35 x - 15 Dividing Polynomials Divide 7x into 28x2. Multiply 4x times 7x+3. Subtract 28x2 + 12x from 28x2 – 23x. Bring down – 15. Divide 7x into –35x. Multiply – 5 times 7x+3. Subtract –35x–15 from –35x–15. Nothing to bring down. So our answer is 4x – 5.

36. - - 10 10 2 2 x x + - + 2 2 x 7 4 x 6 x 8 - 20 x - - 20 x 70 + 8 2 + 14 x 4 x 78 + We write our final answer as + ( 2 x 7 ) Dividing Polynomials Divide 2x into 4x2. Multiply 2x times 2x+7. Subtract 4x2 + 14x from 4x2 – 6x. Bring down 8. Divide 2x into –20x. Multiply -10 times 2x+7. Subtract –20x–70 from –20x+8. 78 Nothing to bring down.