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Multiplying a Polynomial by a Monomial

Multiplying a Polynomial by a Monomial. By: Anna Smoak. Class Background. Algebra 1 Already learned the distributive property and the laws of exponents Goal: Review both of these topics and then have students work together to generalize their knowledge to a more abstract setting.

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Multiplying a Polynomial by a Monomial

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  1. Multiplying a Polynomial by a Monomial By: Anna Smoak

  2. Class Background Algebra 1 Already learned the distributive property and the laws of exponents Goal: Review both of these topics and then have students work together to generalize their knowledge to a more abstract setting

  3. Given: 3x • What does this mean? • Is there another way to write 3x without using the property of multiplication? • 3x = x + x + x • Given: 3(x + 1) • What does this mean? How is this different? • Is there another way to write 3(x+1) without using the property of multiplication? • 3(x + 1) = (x + 1) + (x + 1) + (x + 1) • Now simplify this expression: • 3(x + 1) = (x + 1) + (x + 1) + (x + 1) = x + x + x + 1 + 1 + 1 = 3x + 3

  4. Given: 2(x - 3) • What does this mean? • Is there another way to write 2(x – 3) without using the property of multiplication? • 2(x – 3) = (x – 3) + (x – 3) • Now simplify this expression: • 2(x – 3) = (x – 3) + (x – 3) = x + x – 3 – 3 = 2x – 6 • We have seen that • 3(x + 1) = 3x + 3 • 2(x – 3) = 2x – 6 • What property can we use to get from our original expression to our simplified expression? The Distributive Property

  5. Given -2(x + 1), use the distributive property to simplify the expression. • -2(x + 1) = -2x - 2

  6. Simplify 32 x 34 • How can you write 32 without using exponents? What does it mean for something to be squared? • 32 = 3 x 3 • How can you write 34 without using exponents? What does it mean for something to be raised to the fourth power? • 34 = 3 x 3 x 3 x 3 • Now multiply 32 x 34 using our expanded notation • What property can we use to get from our original expression to our simplified expression? Product of Powers 32 x 34= 36 (3 x 3) x (3 x 3 x 3 x 3) =

  7. What do we do with our coefficients when we multiply two monomials such as (y4)(12y7)? • We multiply our coefficients • What do we do with our variables? • We add their exponents • So what is (y4)(12y7)? • (1 x 12)(y4 + 7)=12y11 • Using the product of powers how would we simplify (4ab6) (7a2b3)? • (4ab6) (7a2b-3) = (4 x 7)(a1 + 2)(b6 +3) = 28a3b9

  8. IN PAIRS Discuss the difference between 3x (2x+ 5) and 3 (2x + 5)

  9. Given 3x2 (2x2 + 5x – 1) discuss in pairs how you think we could simplify this expression. • We must use the distributive property to distribute the 3x2 to each term in (2x2 + 5x – 1). • To perform this distribution we must use the rule of product of powers. • 3x2 (2x2 + 5x – 1) = 3x2(2x2) + 3x2(5x) + 3x2(-1) = 6x4 + 15x3 - 3x2

  10. IN PAIRS • Simplify - (x – 2) + x (6x – 7) • What is our first step? • - (x – 2) + x (6x – 7) • Are we finished now? We must collect like terms • Are we finished now? We must simplify • Are we finished now? We must arrange our terms • Are we finished now? How do you know? = -x + 2 + 6x2 – 7x = -x – 7x + 2 + 6x2 = -8x + 2 + 6x2 = 6x2 – 8x + 2

  11. IN PAIRS Simplify: 6rs(r2s - 3)= 6rs (r2s) + 6rs (-3)= 6r3s2 - 18rs Simplify: -9x (x2 + xy - 2y2)= -9x (x2) - 9x (xy) - 9x (- 2y2)= -9x3 - 9x2y + 18xy2

  12. TICKET OUT OF THE DOOR When a monomial is multiplied by a binomial, will the product always, sometimes, or never be a binomial?

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