5.3 Multiply Polynomials

Before you get started, take this readiness quiz.

Distribute: $2 \left(x + 3\right) .$
If you missed this problem, review the reference.

Simplify: ⓐ $9^{2}$ ⓑ $\left(−9\right)^{2}$ ⓒ $− 9^{2} .$
If you missed this problem, review the reference.

Evaluate: $2 x^{2} - 5 x + 3$ for $x = −2 .$
If you missed this problem, review the reference.

Multiply: ⓐ $\left(3 x^{2}\right) \left(−4 x^{3}\right)$ ⓑ $\left(\frac{5}{6} x^{3} y\right) \left(12 x y^{2}\right) .$

Multiply: ⓐ $\left(5 y^{7}\right) \left(−7 y^{4}\right)$ ⓑ $\left(\frac{2}{5} a^{4} b^{3}\right) \left(15 a b^{3}\right) .$

Multiply: ⓐ $\left(−6 b^{4}\right) \left(−9 b^{5}\right)$ ⓑ $\left(\frac{2}{3} r^{5} s\right) \left(12 r^{6} s^{7}\right) .$

Multiply: ⓐ $−3 y \left(5 y^{2} + 8 y - 7\right)$ ⓑ $4 x^{2} y^{2} \left(3 x^{2} - 5 x y + 3 y^{2}\right) .$

Multiply: ⓐ $4 x^{2} \left(2 x^{2} - 3 x + 5\right)$ ⓑ $−6 a^{3} b \left(3 a^{2} - 2 a b + 6 b^{2}\right) .$

Multiply: ⓐ $\left(x + 8\right) \left(x + 9\right)$ ⓑ $\left(3 c + 4\right) \left(5 c - 2\right) .$

Multiply: ⓐ $\left(5 x + 9\right) \left(4 x + 3\right)$ ⓑ $\left(5 y + 2\right) \left(6 y - 3\right) .$

Multiply: ⓐ $\left(x - 7\right) \left(x + 5\right)$ ⓑ $\left(3 x + 7\right) \left(5 x - 2\right) .$

Multiply: ⓐ $\left(b - 3\right) \left(b + 6\right)$ ⓑ $\left(4 y + 5\right) \left(4 y - 10\right) .$

Multiply: ⓐ $\left(x^{2} + 6\right) \left(x - 8\right)$ ⓑ $\left(2 a b + 5\right) \left(4 a b - 4\right) .$

Multiply: ⓐ $\left(y^{2} + 7\right) \left(y - 9\right)$ ⓑ $\left(2 x y + 3\right) \left(4 x y - 5\right) .$

Multiply using the Vertical Method: $\left(5 m - 7\right) \left(3 m - 6\right) .$

Multiply using the Vertical Method: $\left(6 b - 5\right) \left(7 b - 3\right) .$

Multiply$\left(y - 3\right) \left(y^{2} - 5 y + 2\right)$ using ⓐ the Distributive Property and ⓑ the Vertical Method.

Multiply $\left(x + 4\right) \left(2 x^{2} - 3 x + 5\right)$ using ⓐ the Distributive Property and ⓑ The Vertical Method.

Multiply: ⓐ $\left(x + 9\right)^{2}$ ⓑ $\left(2 c - d\right)^{2} .$

Multiply: ⓐ $\left(y + 11\right)^{2}$ ⓑ $\left(4 x - 5 y\right)^{2} .$

Multiply: ⓐ $\left(6 x + 5\right) \left(6 x - 5\right)$ ⓑ $\left(4 p - 7 q\right) \left(4 p + 7 q\right) .$

Multiply: ⓐ $\left(2 x + 7\right) \left(2 x - 7\right)$ ⓑ $\left(3 x - y\right) \left(3 x + y\right) .$

Choose the appropriate pattern and use it to find the product:

ⓐ $\left(9 b - 2\right) \left(2 b + 9\right)$ ⓑ $\left(9 p - 4\right)^{2}$ ⓒ $\left(7 y + 1\right)^{2}$ ⓓ $\left(4 r - 3\right) \left(4 r + 3\right) .$

Choose the appropriate pattern and use it to find the product:

ⓐ $\left(6 x + 7\right)^{2}$ ⓑ $\left(3 x - 4\right) \left(3 x + 4\right)$ ⓒ $\left(2 x - 5\right) \left(5 x - 2\right)$ ⓓ $\left(6 n - 1\right)^{2} .$

For functions $f \left(x\right) = x + 2$ and $g \left(x\right) = x^{2} - 3 x - 4 ,$ find: ⓐ $\left(f \cdot g\right) \left(x\right)$ ⓑ $\left(f \cdot g\right) \left(2\right) .$

For functions $f \left(x\right) = x - 5$ and $g \left(x\right) = x^{2} - 2 x + 3 ,$ find ⓐ $\left(f \cdot g\right) \left(x\right)$ ⓑ $\left(f \cdot g\right) \left(2\right) .$

For functions $f \left(x\right) = x - 7$ and $g \left(x\right) = x^{2} + 8 x + 4 ,$ find ⓐ $\left(f \cdot g\right) \left(x\right)$ ⓑ $\left(f \cdot g\right) \left(2\right) .$

Multiply the monomials.

ⓐ $\left(−10 x^{5}\right) \left(−3 x^{3}\right)$
ⓑ $\left(\frac{5}{8} x^{3} y\right) \left(24 x^{5} y\right)$

Multiply the monomials.

ⓐ $\left(−6 c^{4}\right) \left(−12 c\right)$
ⓑ $\left(\frac{3}{5} m^{3} n^{2}\right) \left(\frac{5}{9} m^{2} n^{3}\right)$

Multiply.

ⓐ $−5 t \left(t^{2} + 3 t - 18\right) ;$
ⓑ $9 r^{3} s \left(r^{2} - 3 r s + 5 s^{2}\right)$

Multiply.

ⓐ $−5 m \left(m^{2} + 3 m - 18\right)$
ⓑ $−3 x^{2} y^{2} \left(7 x^{2} + 10 x y - y^{2}\right)$

Multiply the binomials using ⓐ the Distributive Property; ⓑ the FOIL method; ⓒ the Vertical Method.

$\left(y + 9\right) \left(y + 3\right)$

Multiply the binomials using ⓐ the Distributive Property; ⓑ the FOIL method; ⓒ the Vertical Method.

$\left(7 q + 4\right) \left(3 q - 8\right)$

Multiply the binomials. Use any method.

$\left(y - 6\right) \left(y - 2\right)$

Multiply the binomials. Use any method.

$\left(6 p + 5\right) \left(p + 1\right)$

Multiply the binomials. Use any method.

$\left(m + 11\right) \left(m - 4\right)$

Multiply the binomials. Use any method.

$\left(3 r - 8\right) \left(11 r + 1\right)$

Multiply the binomials. Use any method.

$\left(y^{2} - 4\right) \left(y + 3\right)$

Multiply the binomials. Use any method.

$\left(2 x y + 3\right) \left(3 x y + 2\right)$

Multiply the binomials. Use any method.

$\left(y^{2} - 7\right) \left(y^{2} - 4\right)$

Multiply the binomials. Use any method.

$\left(3 r s - 7\right) \left(3 r s - 4\right)$

Multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.

$\left(u + 4\right) \left(u^{2} + 3 u + 2\right)$

Multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.

$\left(a + 10\right) \left(3 a^{2} + a - 5\right)$

Multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.

$\left(2 a^{2} - 5 a + 10\right) \left(3 a^{2} + a - 5\right)$

Multiply. Use either method.

$\left(p - 4\right) \left(p^{2} - 6 p + 9\right)$

Multiply. Use either method.

$\left(6 r + 1\right) \left(r^{2} - 7 r - 9\right)$

Square each binomial using the Binomial Squares Pattern.

$\left(q + 12\right)^{2}$

Square each binomial using the Binomial Squares Pattern.

$\left(2 y - 3 z\right)^{2}$

Square each binomial using the Binomial Squares Pattern.

$\left(x + \frac{2}{3}\right)^{2}$

Square each binomial using the Binomial Squares Pattern.

$\left(\frac{1}{8} x - \frac{1}{9} y\right)^{2}$

Square each binomial using the Binomial Squares Pattern.

$\left(5 u^{2} + 9\right)^{2}$

Square each binomial using the Binomial Squares Pattern.

$\left(8 p^{3} - 3\right)^{2}$

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(8 j + 4\right) \left(8 j - 4\right)$

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(9 c + 5\right) \left(9 c - 5\right)$

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(7 w + 10 x\right) \left(7 w - 10 x\right)$

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(p + \frac{4}{5} q\right) \left(p - \frac{4}{5} q\right)$

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(x y - 9\right) \left(x y + 9\right)$

Multiply each pair of conjugates using the Product of Conjugates Pattern.

$\left(15 m^{2} - 8 n^{4}\right) \left(15 m^{2} + 8 n^{4}\right)$

Find each product.

$\left(t - 9\right)^{2}$

Find each product.

$\left(2 x + y\right) \left(x - 2 y\right)$

Find each product.

$\left(3 p + 8\right) \left(3 p - 8\right)$

Find each product.

$\left(k - 6\right)^{2}$

Find each product.

$\left(x^{2} + 8 y\right) \left(8 x - y^{2}\right)$

Find each product.

$\left(y^{4} + 2 z\right)^{2}$

Find each product.

$\left(m^{3} - 8 n\right)^{2}$

Find each product.

$\left(r^{2} - s^{3}\right) \left(r^{3} + s^{2}\right)$

$\left(15 p - 4\right) + \left(3 p - 5\right)$

$\left(j^{2} - 8 j - 27\right) - \left(j^{2} + 2 j - 12\right)$

$\left(\frac{1}{4} d^{5}\right) \left(36 d^{2}\right)$

$\left(6 m^{4} n^{3}\right) \left(7 m n^{5}\right)$

$5 q^{3} \left(q^{2} - 2 q + 6\right)$

$\left(y^{2} - 2 y\right) \left(y + 1\right)$

$\left(6 k - 1\right) \left(k^{2} + 2 k - 4\right)$

$\left(11 - b\right) \left(11 + b\right)$

$\left(2 x^{2} - 3 y^{4}\right) \left(2 x^{2} + 3 y^{4}\right)$

$\left(3 d + 1\right)^{2}$

$\left(3 z + \frac{1}{5}\right)^{2}$

Multiply Polynomial Functions

For functions $f \left(x\right) = x - 1$ and $g \left(x\right) = 4 x^{2} + 3 x - 5 ,$ find ⓐ $\left(f \cdot g\right) \left(x\right)$ ⓑ $\left(f \cdot g\right) \left(−2\right)$

Multiply Polynomial Functions

For functions $f \left(x\right) = 7 x - 8$ and $g \left(x\right) = 7 x + 8 ,$ find ⓐ $\left(f \cdot g\right) \left(x\right)$ ⓑ $\left(f \cdot g\right) \left(−2\right)$

Multiply Polynomial Functions

For functions $f \left(x\right) = x^{2} + 4 x - 3$ and $g \left(x\right) = x^{2} + 2 x + 4 ,$ find ⓐ $\left(f \cdot g\right) \left(x\right)$ ⓑ $\left(f \cdot g\right) \left(1\right)$

Multiply the following:

$\begin{matrix}\left(x + 2\right) \left(x - 2\right) \\ \left(y + 7\right) \left(y - 7\right) \\ \left(w + 5\right) \left(w - 5\right)\end{matrix}$

Explain the pattern that you see in your answers.

Why does $\left(a + b\right)^{2}$ result in a trinomial, but $\left(a - b\right) \left(a + b\right)$ result in a binomial?