EDGE Math 8

Multiplication of Polynomials

You learned that when multiplying polynomials, you apply various properties of real numbers such as the distributive, associative, and commutative properties, as well as the laws of exponents.

Example 1

Multiply $3 x (2 x y + 5 x^{3} y - x y^{2}) .$

Solution

Apply the distributive property; that is, multiply the monomial $3 x$ by every term of the polynomial $2 x y + 5 x^{3} y - x y^{2} .$

3 x (2 x y + 5 x^{3} y - x y^{2})

= 3 x (2 x y) + 3 x (5 x^{3} y)

+ 3 x (- x y^{2})

= 6 x^{2} y + 15 x^{4} y - 3 x^{2} y^{2}

Example 2

Multiply $(4 x^{2} - 7 x + 2) (3 x + 4) .$

Solution

Multiply every term of the polynomial $4 x^{2} - 7 x + 2$ by every term of the binomial $3 x + 4.$ Then combine like terms.

(4 x^{2} - 7 x + 2) (3 x + 4)

= 4 x^{2} (3 x) + 4 x^{2} (4) - 7 x (3 x)

- 7 x (4) + 2 (3 x) + 2 (4)

= 12 x^{3} + 16 x^{2} - 21 x^{2} - 28 x

+ 6 x + 8

= 12 x^{3} - 5 x^{2} - 22 x + 8