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Learn how to multiply polynomials and expand binomial expressions raised to positive integer powers. Explore real-world business applications and practice expanding powers of binomials.
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Essential Questions • How do we multiply polynomials? • How do we use binomial expansion to expand binomial expressions that are raised to positive integer powers?
Example 1: Business Application A standard Burly Box is p ft by 3p ft by 4p ft. A large Burly Box has 1.5 ft added to each dimension. Write a polynomial V(p) in standard form that can be used to find the volume of a large Burly Box. The volume of a large Burly Box is the product of the area of the base and height. The area of the base of the large Burly Box is the product of the length and width of the box. The length, width, and height of the large Burly Box are greater than that of the standard Burly Box.
Example 1: Business Application Solve A(p) = l(p) w(p). Solve V(p) = A(p) h(p). p + 1.5 3p2+ 6p + 2.25 3p+ 1.5 4p+ 1.5 The volume of a large Burly Box can be modeled byV(p) = 12p3+ 28.5p2 + 18p + 3.375
Example 2: Business Application Mr. Silva manages a manufacturing plant. From 1990 through 2005 the number of units produced (in thousands) can be modeled by N(x) = 0.02x2 + 0.2x + 3. The average cost per unit (in dollars) can be modeled by C(x) = –0.004x2 – 0.1x + 3. Write a polynomial T(x) that can be used to model the total costs. Total cost is the product of the number of units and the cost per unit. Multiply the two polynomials.
Example 2: Business Application 0.02x2+ 0.2x + 3 –0.004x2– 0.1x+ 3 Mr. Silva’s total manufacturing costs, in thousands of dollars, can be modeled by T(x) = –0.00008x4– 0.0028x3 + 0.028x2+ 0.3x + 9
Example 3: Expanding a Power of a Binomial Find the product. (a + 2b)3 Write in expanded form. (a + 2b)(a + 2b)(a + 2b) Multiply the last two binomial factors. (a + 2b)(a2 + 4ab + 4b2) Distribute a and then 2b. Combine like terms.
Example 4: Expanding a Power of a Binomial Find the product. (x + 4)4 (x + 4)(x + 4)(x + 4)(x + 4) Write in expanded form. Multiply the last two binomial factors. (x + 4)(x + 4)(x2 + 8x + 16) Multiply the first two binomial factors. (x2 + 8x + 16)(x2 + 8x + 16) Distribute x2 and then 8x and then 16. Combine like terms.
Example 5: Expanding a Power of a Binomial Find the product. (2x – 1)3 (2x – 1)(2x – 1)(2x – 1) Write in expanded form. (2x – 1)(4x2 – 4x + 1) Multiply the last two binomial factors. Distribute 2x and then –1. Combine like terms.
