1 Topic Polynomial Multiplication

2 Lesson California Standards: 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. What it means for you: You’ll learn how to multiply monomials and polynomials. Polynomial Multiplication Topic Key words: polynomial monomial distributive property degree

3 Lesson To multiply two polynomials together, you have to multiply every single term together, one by one. Polynomial Multiplication Topic x ( y + 4) = ( x × y ) + ( x × 4)

4 Lesson The Distributive Property and Polynomial Products Polynomial Multiplication Topic In Topic you saw the method for multiplying a polynomial by a number — you multiply each term separately by that number. This method’s based on the distributive property from Topic In the same sort of way, when you multiply a polynomial by a monomial, you multiply each term separately by that monomial — again, using the distributive property.

5 Polynomial Multiplication Example 1 Topic Simplify the expression –2 a ( a + 3 a 2 ). Solution –2 a ( a + 3 a 2 ) is a product of the monomial –2 a and the binomial ( a + 3 a 2 ), so multiply each term of the binomial by the monomial. Solution follows… = –2 a ( a ) + (–2 a )(3 a 2 ) = –2 a 2 – 6 a 3

6 Lesson The Distributive Property and Polynomial Products Polynomial Multiplication Topic To find the product of two polynomials, such as ( a – 2 b )(3 a + b ), you use the distributive property twice. The Distributive Property a ( b + c ) = ab + ac

7 ( a – 2 b )(3 a + b ) Polynomial Multiplication Example 2 Topic Simplify the expression ( a – 2 b )(3 a + b ). Solution Solution follows… Use the distributive property twice = 3 a 2 – 5 ab – 2 b 2 = 3 a 2 + ab – 6 ab – 2 b 2 = (3 a 2 + ab ) + (–6 ab – 2 b 2 ) = [( a )(3 a ) + ( a )( b )] + [(–2 b )(3 a ) + (–2 b )( b )] = ( a )(3 a + b ) + (–2 b )(3 a + b )

8 (3 x – 2 m )(4 x – 3 m ) Polynomial Multiplication Example 3 Topic Simplify (3 x – 2 m )(4 x – 3 m ). Solution Solution follows… = 12 x 2 – 17 mx + 6 m 2 = 12 x 2 – 9 mx – 8 mx + 6 m 2 = 3 x (4 x – 3 m ) – 2 m (4 x – 3 m )

9 ( v + 3)(4 + v ) Polynomial Multiplication Example 4 Topic Simplify ( v + 3)(4 + v ). Solution Solution follows… = v v + 12 = 4 v + v v = v (4 + v ) + 3(4 + v )

10 Expand and simplify each product, using the distributive method. Show all your work. Lesson Guided Practice Polynomial Multiplication Topic Solution follows… 1. ( m + c )( m + 2 c )2. ( x – 3 y )( x + 2 y ) 3. (2 x – 3)(2 x + 5)4. ( a – 4 b )( a + 3 b ) 5. (3 x – 5)(2 x – 3)6. (5 x + 3 y )(2 x + 3 y ) = m ( m + 2 c ) + c ( m + 2 c ) = m mc + mc + 2 c 2 m mc + 2 c 2 = x ( x + 2 y ) + (–3 y )( x + 2 y ) = x xy – 3 xy – 6 y 2 = x 2 – xy – 6 y 2 = 2 x (2 x + 5) + (–3)(2 x + 5) = 4 x x – 6 x – 15 = 4 x x – 15 = a ( a + 3 b ) + (–4 b )( a + 3 b ) = a ab – 4 ab – 12 b 2 = a 2 – ab – 12 b 2 = 3 x (2 x – 3) + (–5)(2 x – 3) = 6 x 2 – 9 x – 10 x + 15 = 6 x 2 – 19 x + 15 = 5 x (2 x + 3 y ) + 3 y (2 x + 3 y ) = 10 x xy + 6 xy + 9 y 2 = 10 x xy + 9 y 2

11 Lesson Guided Practice Polynomial Multiplication Topic Solution follows… Determine whether the following are correct for the products given. 7. ( a + b )( a – b ) = a 2 – b 2 8. ( a + b ) 2 = a 2 + b 2 9. ( a – b ) 2 = a 2 – 2 ab + b ( a + b )( a + b ) = a ab + b 2 Correct Not correct

12 Topic Polynomial Multiplication You Can Multiply Polynomials with Lots of Terms It doesn’t matter how many terms are in the polynomials that you’re multiplying — the method is just the same. You have to multiply each term in one set of parentheses by every term in the second set of parentheses.

13 Example 5 Solution follows… Topic Polynomial Multiplication Simplify ( x + 2)( x x + 3). Solution ( x + 2)( x x + 3) = x x x + 6 = x x x + 2 x x + 6 = x ( x x + 3) + 2( x x + 3)

14 Topic Polynomial Multiplication The Highest Power Gives the Degree of a Polynomial The degree of a polynomial in x is the size of the highest power of x in the expression. For example, a third-degree polynomial will contain at least one x 3 term, but won’t contain x 4 or any higher powers of x.

15 ( x – 3)(2 x 2 – 3 x + 2) Example 6 Solution follows… Topic Polynomial Multiplication Simplify ( x – 3)(2 x 2 – 3 x + 2) and state the degree of the product. Solution The term 2 x 3 has the highest power, so the degree is 3. = x (2 x 2 – 3 x + 2) – 3(2 x 2 – 3 x + 2) = 2 x 3 – 3 x x – 6 x x – 6 = 2 x 3 – 9 x x – 6

16 Lesson Guided Practice Polynomial Multiplication Topic Solution follows… Expand and simplify each product, and state the degree of the resulting polynomial. 11. ( x + 3)(2 x 2 – 3 x + 1) 12. (2 y – 3)(–3 y 2 – y + 1) 13. ( x 2 – 3 x + 4)(2 x + 1) 14. (3 y y – 2)(4 y – 1) 15. (3 x + 4)(–2 x 2 + x – 2) 16. (2 x – 3) 2 –6 x 3 – 5 x 2 – 2 x – 8, 3rd degree4 x 2 – 12 x + 9, 2nd degree 12 y 4 – 3 y y 2 – 12 y + 2, 4th degree2 x 3 – 5 x x + 4, 3rd degree –6 y y y – 3, 3rd degree2 x x 2 – 8 x + 3, 3rd degree

17 Lesson Guided Practice Polynomial Multiplication Topic Solution follows… Determine whether the following are correct for the products given. 17. ( a 2 + b 2 )( a – ab + b ) = a 3 + b ( a + b )( a 2 – ab + b 2 ) = a 3 + b ( a – b )( a 2 + ab + b 2 ) = a 3 – b ( a 2 – b 2 )( a + ab + b ) = a 3 – b 3 Correct Not correct Correct Not correct

18 Lesson You Can Also Use the Stacking Method Polynomial Multiplication Topic You can find the product of 63 and 27 by “stacking” the two numbers and doing long multiplication: You can use the same idea to find the products of polynomials — just make sure you keep like terms in the same columns. 7 × 63 2 × ×

19 Polynomial Multiplication Example 7 Topic Expand and simplify the product (2 x + 3 y )( x + 5 y ). Solution Solution follows… 5 y (2 x + 3 y ) 2 x + 13 y 2 × 10 yx + 15 y 2 10 xy + 15 y x xy + 15 y 2 2 x xy + 15 y 2 x (2 x + 3 y )

20 Polynomial Multiplication Example 8 Topic Simplify ( x – 2)(2 x 2 – 3 x + 4). Solution Solution follows… –2(2 x 2 – 3 x + 4) 2 x 2 – 13 x + 4 × 10 yx – 2 –4 x x – x 3 – 3 x x x 3 – 7 x x – 8 x (2 x 2 – 3 x + 4)

21 Lesson Guided Practice Polynomial Multiplication Topic Solution follows… Use the stacking method to multiply these polynomials: 21. (3 x + y )( x + 2 y ) 22. (4 x + 5 y )(2 x + 3 y ) 23. (3 x x + 3)(3 x – 4) 24. (4 x 2 – 5 x + 6)(4 x + 5) 3 xy + 2 y 2 × xy + 2 y 2 6 xy + 2 y x xy + 2 y 2 3 x xy + 2 y 2 4 xy + 15 y 2 × 2 xy + 13 y 2 12 xy + 15 y x xy + 12 y 2 8 x xy + 15 y 2 3 x x + 13 × 3 x – 14 –12 x 2 – 8 x – x x x x 3 – 6 x x – 12 4 x 2 – 5 x + 16 × 4 x x 2 – 25 x x 3 – 20 x x x 3 – 20 x 2 – 24 x + 30

22 Lesson Guided Practice Polynomial Multiplication Topic Solution follows… Use the stacking method to multiply these polynomials: 25. ( a + b ) ( a – b ) ( a – b )( a + b )28. ( a – b )( a 2 + ab + b 2 ) ay + b 2 × ay + b 2 ab + b 2 + a ab + y 2 a ab + b 2 ay – b 2 × ay – b 2 – ab + b 2 + a 2 – 7 ab + y 2 a 2 – 2 ab + b 2 ay – b 2 × ay + b 2 ab – b 2 + a 2 – ab + y 2 a 2 + ab – b 2 a 2 + ab + b 2 × a b – b 2 – a 2 b – ab 2 – b 3 + a 3 + a 2 b + ab 2 + b 3 a 3 – a 2 b + ab 2 – b 3

23 Lesson Guided Practice Polynomial Multiplication Topic Solution follows… Use the stacking method to multiply these polynomials: 29. ( a + b )( a 2 – ab + b 2 )30. ( a 2 – b 2 )( a 2 + b 2 ) a 2 – ab + b 2 × a b + b 2 a 2 b – ab 2 + b 3 + a 3 – a 2 b + ab 2 + b 3 a 3 – a 2 b + ab 2 + b 3 a 2 + b 2 × a 2 – b 2 – a 2 b 2 – b 4 + a 4 + a 2 b 2 + b 4 a 4 + a 2 b 2 – b 4

24 Expand and simplify each product, using the distributive method. Show all your work. Polynomial Multiplication Independent Practice Solution follows… Topic (2 x + 8)( x – 4) 2. ( x 2 + 3)( x – 2) 3. ( x – 3)(2 – x ) 4. (2 x + 7)(3 x + 5) 5. (3 x – 8)( x 2 – 4 x + 2) 6. (2 x – 4 y )(3 x – 3 y + 4) = 2 x ( x – 4) + 8( x – 4) = 2 x 2 – 32 = x 2 ( x – 2) + 3( x – 2) = x 3 – 2 x x – 6 = x (2 – x ) + (–3)(2 – x ) = – x x – 6 = 2 x (3 x + 5) + 7(3 x + 5) = 6 x x + 35 = 3 x ( x 2 – 4 x + 2) + (–8)( x 2 – 4 x + 2) = 3 x 3 – 20 x x – 16 = 2 x (3 x – 3 y + 4) + (–4 y )(3 x – 3 y + 4) = 6 x 2 – 18 xy + 8 x + 12 y 2 – 16 y

25 Polynomial Multiplication Independent Practice Solution follows… Topic Use the stack method to multiply. Show all your work. 7. ( x 2 – 4)( x + 3) 8. ( x – y )(3 x 2 + xy + y 2 ) 9. (4 x 2 – 5 x )(1 + 2 x – 3 x 2 ) 10. ( x + 4)(3 x 2 – 2 x + 5) x x – 14 × x x 2 – 8 x – 12 + x x 2 – 4 x + 12 x x 2 – 4 x – 12 3 x 2 + xy + y 2 × x – y –3 x 2 y – xy 2 – y x x 2 y + xy 2 – y 3 3 x 3 – 2 x 2 y + xy 2 – y 3 –3 x x + 1 × 4 x 2 – 5 x x 3 – 10 x 2 – 5 x –12 x x x 2 – 5 x + 1 –12 x x 3 – 16 x 2 – 5 x x 2 – 2 x + 15 × x x 2 – 8 x – x 3 – 12 x x – 20 3 x x 2 – 3 x – 20

26 Use these formulas to find each of the products in Exercises 11–16. Polynomial Multiplication Independent Practice Solution follows… Topic ( a + b ) 2 = a ab + b 2 ( a – b ) 2 = a 2 – 2 ab + b 2 ( a + b )( a – b ) = a 2 – b ( x + 2) (3 x – 1)(3 x + 1) 13. (2 x – 3) (4 x + y ) (5 x + 3 c )(5 x – 3 c )16. (8 c + 3) 2 x x x 2 – 1 4 x 2 – 12 x x xy + y 2 25 x 2 – 9 c 2 64 c c + 9

27 Topic Round Up Polynomial Multiplication Watch out for the signs when you’re subtracting polynomials. It’s usually a good idea to put parentheses around the polynomial you’re subtracting, to make it easier to keep track of the signs.