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# Multiplying a Polynomial by a Monomial

To multiply a monomial by a polynomial with more than one term, use the Distributive Property to distribute the monomial to each term in the polynomial.

Example 1

Find: -8w3y(4w2y5 - w4)

 Solution -8w3y(4w2y5 - w4) Multiply each term in the polynomial by the monomial, -8w3y. = (-8w3y)(4w2y5) - (-8w3y)(w4) Within each term, write the coefficients next to each other. Write the factors with base w next to each other and write the factors with base y next to each other. = (-8 Â· 4)(w3 Â· w2)(y Â· y5) - (-8)(w3 Â· w4)(y) Use the Multiplication Property of Exponents. = (-8 Â· 4)(w3 + 2 y1 + 5) - (-8)(w3 + 4 y) Simplify. = -32w5y6 + 8w7y

Example 2

Find: 5x4(3x2y2 - 2xy2 + x3y)

 Solution 5x4(3x2y2 - 2xy2 + x3y) Multiply each term in the polynomial by the monomial, 5x4. = (5x4)(3x2y2) - (5x4)(2xy2) + (5x4)(x3y) Within each term, write the coefficients next to each other. Write the factors with base x next to each other and write the factors with base y next to each other. = (5 Â· 3)(x4 x 2 y2) - (5 Â· 2)(x4 x 1y2) + (5 Â· 1)(x4 x 3 y) Use the Multiplication Property of Exponents. = (5 Â· 3)(x4 + 2 y2) - (5 Â· 2)(x4 + 1 y2) + (5 Â· 1)(x4 + 3 y) Simplify. = 15x6y2 - 10x5y2 + 5x7y