7.2 Multiplying Polynomial Expressions and Functions

Monomials, Binomials, and Trinomials

Product Property for Exponents If n and m are counting numbers, then xmxn = xm + n In words: To multiply two powers of x, keep the base and add the exponents.

Example: Finding the Product of Two Monomials Find the product.

Solution

Example: Finding the Product of a Monomial and a Polynomial Find the product. 1. 2. 3.

Solution

Solution

Multiplying Two Polynomials To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial. Then combine like terms if possible.

Example: Finding Products of Binomials Find the product. 1. 2. 3. 4.

Solution

Solution

Example: Finding the Products of Two Polynomials Find the product. 1. (3a – 2)(a2 + 4a + 5) 2. (2x + y)(5x2 – 3xy + 4y2)

Solution 1. To begin, multiply each term in the first polynomial by each term in the second polynomial:

Solution 2. Multiply each term in the first polynomial by each term in the second polynomial:

Product Function Definition If f and g are functions and x is in the domain of both functions, then we can form the product function f ∙ g: (f ∙ g)(x) = f(x) ∙ g(x)

Example: Using a Product Function to Model a Situation The annual cost of state corrections (prisons and related costs) per person in the United States can be modeled by the function C(t) = 3.3t + 89, where C(t) is the annual cost (in dollars per person at t years since 1990. The U.S. population can be modeled by the function P(t) = 2.8t + 254, where P(t) is the population (in millions) at t years since 1990. Data is shown in the table on the next slide.

Example: Using a Product Function to Model a Situation

Example: Using a Product Function to Model a Situation 1. Check that the models fit the data well. 2. Find an equation of the product function C ∙ P. 3. Perform a unit analysis of the expression C(t) ∙ P(t). 4. Find (C ∙ P)(28). What does it mean in this situation? 5. Use a graphing calculator graph to determine whether the function C ∙ P is increasing, decreasing, or neither for values of t between 0 and 30. What does your result mean in this situation?

Solution 1. Check the fit of the cost model on the left and the fit of the population model on the right. The models appear to fit the data fairly well.

Solution 2. 3. The units of the expressions are millions of dollars.

Solution 4. This means the total cost of state corrections will be about $60,297 million ($60.297 billion) in 2018, according to the model.

Solution 5. To graph the model, set the window as shown. For values of t between 0 and 30, the model is increasing. This means the total cost of state corrections has been increasing since 1990 and will continue to increase until 2020.