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Understanding Polynomial-Monomial Multiplication and Simplification Techniques

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This guide focuses on multiplying a monomial by a polynomial using horizontal and vertical methods, illustrating the distributive property for clarity. The process is demonstrated through step-by-step examples, including how to simplify complex expressions. Additionally, it covers real-world applications, such as calculating expenses at an amusement park based on ride types. Learn to formulate equations and solve for variables effectively, combining like terms to arrive at the correct answers. Perfect for students looking to strengthen their algebra skills.

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Understanding Polynomial-Monomial Multiplication and Simplification Techniques

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  1. Section 7.5 Multiplying a Polynomial by a Monomial

  2. Find the product of a monomial and a polynomial. • Solve equations involving polynomials.

  3. Multiply a Polynomial by a Monomial Find 6y(4y2 – 9y – 7). Method 1 Horizontal 6y(4y2 – 9y – 7) = 6y(4y2) – 6y(9y) – 6y(7) Distributive Property = 24y3 – 54y2 – 42y Multiply. Answer: 24y3 – 54y2 – 42y

  4. 4y2 – 9y – 7 (x) 6y Distributive Property Multiply a Polynomial by a Monomial Method 2 Vertical 24y3 – 54y2 – 42y Multiply. Answer: 24y3 – 54y2 – 42y

  5. Simplify Expressions Simplify 3(2t2 – 4t – 15) + 6t(5t + 2). 3(2t2 – 4t – 15) + 6t(5t + 2) = 3(2t2) – 3(4t) – 3(15) + 6t(5t) + 6t(2) Distributive Property = 6t2 – 12t – 45 + 30t2 + 12t Product of Powers = (6t2 + 30t2) + [(– 12t) + 12t]– 45 Commutative and Associative Properties Answer: = 36t2 – 45 Combine like terms.

  6. Amountof money $3 perride regularrides superrides $2 per ride. admission + + =   A. ENTERTAINMENTAdmission to the Super Fun Amusement Park is $10. Once in the park, super rides are an additional $3 each and regular rides are an additional $2. Sarita goes to the park and rides 15 rides, of which s of those 15 are super rides. Find an expression for how much money Sarita spent at the park. Words VariableIf s = the number of super rides, then15 – s is the number of regular rides. Let M be the amount of money Sarita spent at the park.

  7. Amountof money $3 perride regularrides superrides $2 per ride. admission + + =   Equation M = 10 + s● 3 + (15 – s) ● 2 = 10 +3s + 15(2) – s(2) Distributive Property = 10 + 3s + 30 – 2s Simplify. = 40 + s Simplify. Answer: An expression for the amount of money Saritaspent in the park is 40 + s, where s is the number of super rides she rode.

  8. B. Evaluate the expression to find the cost if Sarita rode 9 super rides. 40 + s = 40 + 9 s = 9 = 49 Add. Answer:Sarita spent $49.

  9. Polynomials on Both Sides Solve b(12 + b) – 7 = 2b + b(–4 + b). b(12 + b) – 7 = 2b + b(–4 + b) Original equation 12b + b2 – 7 = 2b – 4b + b2 Distributive Property 12b + b2 – 7 = –2b + b2 Combine like terms. 12b – 7 = –2b Subtract b2 from each side.

  10. Polynomials on Both Sides 12b = –2b + 7 Add 7 to each side. 14b = 7 Add 2b to each side. Divide each side by 14. Answer:

  11. Homework Assignment #29 7.5 Skills Practice Sheet

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