10.6 Introduction to Factoring Polynomials

Before you get started, take this readiness quiz.

Factor $56$ into primes.
If you missed this problem, review the reference.

Multiply: $−3 \left(6 a + 11\right) .$
If you missed this problem, review the reference.

Multiply: $4 x^{2} \left(x^{2} + 3 x - 1\right) .$
If you missed this problem, review the reference.

Find the greatest common factor: $54 , 36 .$

Find the greatest common factor: $48 , 80 .$

Find the greatest common factor: $7 y , 14 .$

Find the greatest common factor: $22 , 11 m .$

Find the greatest common factor: $16 x^{2} , 24 x^{3} .$

Find the greatest common factor: $27 y^{3} , 18 y^{4} .$

Find the greatest common factor: $21 x^{3} , 9 x^{2} , 15 x .$

Find the greatest common factor: $25 m^{4} , 35 m^{3} , 20 m^{2} .$

Factor: $4 x + 12 .$

Factor: $6 a + 24 .$

Factor: $9 a + 9 .$

Factor: $11 x + 11 .$

Factor: $11 x - 44 .$

Factor: $13 y - 52 .$

Factor: $4 y^{2} + 8 y + 12 .$

Factor: $6 x^{2} + 42 x - 12 .$

Factor: $9 x^{2} + 7 x .$

Factor: $5 a^{2} - 12 a .$

Factor: $2 x^{3} + 12 x^{2} .$

Factor: $6 y^{3} - 15 y^{2} .$

Factor: $18 y^{2} + 63 y .$

Factor: $32 k^{2} + 56 k .$

Factor: $18 y^{3} - 6 y^{2} - 24 y .$

Factor: $16 x^{3} + 8 x^{2} - 12 x .$

Factor: $−5 y - 35 .$

Factor: $−16 z - 56 .$

Factor: $−7 a^{2} + 21 a .$

Factor: $−6 x^{2} + x .$

Find the greatest common factor.

$45 , 75$

Find the greatest common factor.

$150 , 275$

Find the greatest common factor.

$4 y , 28$

Find the greatest common factor.

$5 b , 30$

Find the greatest common factor.

$9 x , 15 x^{2}$

Find the greatest common factor.

$12 p^{4} , 48 p^{3}$

Find the greatest common factor.

$18 a , 6 a^{2} , 22 a^{3}$

Find the greatest common factor.

$40 y , 10 y^{2} , 90 y^{3}$

Find the greatest common factor.

$35 x^{3} , 10 x^{4} , 5 x^{5}$

Find the greatest common factor.

$14 b^{2} , 35 b^{3} , 63 b^{4}$

Factor the greatest common factor from each polynomial.

$5 y + 15$

Factor the greatest common factor from each polynomial.

$4 b - 20$

Factor the greatest common factor from each polynomial.

$7 x - 7$

Factor the greatest common factor from each polynomial.

$3 n^{2} + 21 n + 12$

Factor the greatest common factor from each polynomial.

$6 q^{2} + 30 q + 42$

Factor the greatest common factor from each polynomial.

$9 c^{2} + 22 c$

Factor the greatest common factor from each polynomial.

$17 x^{2} + 7 x$

Factor the greatest common factor from each polynomial.

$4 q^{2} + 7 q$

Factor the greatest common factor from each polynomial.

$3 r^{2} + 27 r$

Factor the greatest common factor from each polynomial.

$30 u^{2} - 10 u$

Factor the greatest common factor from each polynomial.

$a b + 8 b$

Factor the greatest common factor from each polynomial.

$55 y - 11 y^{4}$

Factor the greatest common factor from each polynomial.

$45 c^{3} - 15 c^{2}$

Factor the greatest common factor from each polynomial.

$6 c^{3} - 6 c d^{2}$

Factor the greatest common factor from each polynomial.

$48 x^{3} + 72 x^{2}$

Factor the greatest common factor from each polynomial.

$144 a^{6} + 90 a^{3}$

Factor the greatest common factor from each polynomial.

$10 y^{2} + 50 y + 40$

Factor the greatest common factor from each polynomial.

$12 u^{2} - 36 u - 108$

Factor the greatest common factor from each polynomial.

$5 p^{4} - 20 p^{3} - 15 p^{2}$

Factor the greatest common factor from each polynomial.

$8 c^{5} + 40 c^{4} - 56 c^{3}$

Factor the greatest common factor from each polynomial.

$−7 p - 84$

Factor the greatest common factor from each polynomial.

$−18 b^{2} - 66 b$

Factor the greatest common factor from each polynomial.

$−8 a^{3} + 32 a^{2}$

Factor the greatest common factor from each polynomial.

$−9 b^{5} + 63 b^{3}$

Height of a baseball The height of a baseball hit with velocity $80$ feet/second at $4$ feet above ground level is $−16 t^{2} + 80 t + 4 ,$ with $t =$ the number of seconds since it was hit. Factor the greatest common factor from this polynomial.

What is the GCF of $y^{4}$, $y^{5}$, and $y^{10}$? Write a general rule that tells how to find the GCF of $y^{\text{a}}$, $y^{\text{b}}$, and $y^{\text{c}}$.

Determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

$y^{2} + 8 y - 20$

Determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

$9 x^{3} - 1$

Determine the degree of each polynomial.

$16 x^{2} - 40 x - 25$

Determine the degree of each polynomial.

$−15$

Add or subtract the monomials.

$4 p + 11 p$

Add or subtract the monomials.

Add $4 n^{5} , − n^{5} , −6 n^{5}$

Add or subtract the polynomials.

$\left(4 a^{2} + 9 a - 11\right) + \left(6 a^{2} - 5 a + 10\right)$

Add or subtract the polynomials.

$\left(y^{2} - 3 y + 12\right) + \left(5 y^{2} - 9\right)$

Add or subtract the polynomials.

Find the sum of $8 q^{3} - 27$ and $q^{2} + 6 q - 2$

Evaluate each polynomial for the given value.

$200 x - \frac{1}{5} x^{2}$ when $x = 5$

Evaluate each polynomial for the given value.

$200 x - \frac{1}{5} x^{2}$ when $x = 15$

Evaluate each polynomial for the given value.

$5 + 40 x - \frac{1}{2} x^{2}$ when $x = −4$

Evaluate each polynomial for the given value.

A pair of glasses is dropped off a bridge $640$ feet above a river. The polynomial $−16 t^{2} + 640$ gives the height of the glasses $t$ seconds after they were dropped. Find the height of the glasses when $t = 6 .$

Simplify.

$6^{3}$

Simplify.

$\left(−0.5\right)^{2}$

Simplify each expression.

$p^{3} \cdot p^{10}$

Simplify each expression.

$a \cdot a^{2} \cdot a^{3}$

Simplify each expression.

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

Simplify each expression.

$\left(3^{2}\right)^{5}$

Simplify each expression.

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

Simplify each expression.

$\left(2 a b\right)^{8}$

Simplify each expression.

$\left(3 a^{5}\right)^{3}$

Simplify each expression.

$\left(x^{3}\right)^{5} \left(x^{2}\right)^{3}$

Multiply the monomials.

$\left(−6 p^{4}\right) \left(9 p\right)$

Multiply the monomials.

$\left(8 x^{2} y^{5}\right) \left(7 x y^{6}\right)$

Multiply.

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

Multiply.

$−5 y \left(125 y^{3} - 1\right)$

Multiply the binomials using various methods.

$\left(a + 5\right) \left(a + 2\right)$

Multiply the binomials using various methods.

$\left(3 x + 1\right) \left(2 x - 7\right)$

Multiply the binomials using various methods.

$\left(n + 8\right) \left(n + 1\right)$

Multiply the binomials using various methods.

$\left(5 u - 3\right) \left(u + 8\right)$

Multiply the binomials using various methods.

$\left(p + 4\right) \left(p + 7\right)$

Multiply the binomials using various methods.

$\left(3 c + 1\right) \left(9 c - 4\right)$

Multiply using any method.

$\left(x + 1\right) \left(x^{2} - 3 x - 21\right)$

Multiply using any method.

$\left(m + 6\right) \left(m^{2} - 7 m - 30\right)$

Simplify.

$\frac{2^{8}}{2^{2}}$

Simplify.

$\frac{n^{3}}{n^{12}}$

Simplify.

$3^{0}$

Simplify.

$\left(14 t\right)^{0}$

Simplify.

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

Simplify.

$\left(\frac{5 m}{n}\right)^{3}$

Simplify.

$\frac{\left(a^{3}\right)^{2}}{a^{4}}$

Simplify.

$\left(\frac{x}{x^{9}}\right)^{5}$

Simplify.

$\frac{\left(n^{5}\right)^{3}}{\left(n^{2}\right)^{8}}$

Divide the monomials.

$72 p^{12} \div 8 p^{3}$

Divide the monomials.

$\frac{ 45 y^{6}}{−15 y^{10}}$

Divide the monomials.

$\frac{28 a^{9} b}{7 a^{4} b^{3}}$

Divide the monomials.

$\frac{\left(5 m^{9} n^{3}\right) \left(8 m^{3} n^{2}\right)}{\left(10 m n^{4}\right) \left(m^{2} n^{5}\right)}$

Simplify.

$6^{−2}$

Simplify.

$5 \cdot 2^{−4}$

Simplify.

$x^{−3} \cdot x^{9}$

Simplify.

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

Simplify.

$\left(k^{−2}\right)^{−3}$

Simplify.

$\frac{b^{8}}{ b^{−2}}$

Write each number in scientific notation.

$5,300,000$

Write each number in scientific notation.

The thickness of a piece of paper is about $0.097$ millimeter.

Convert each number to decimal form.

$2.9 \times 10^{4}$

Convert each number to decimal form.

$3.75 \times 10^{−1}$

Multiply and write your answer in decimal form.

$\left(3 \times 10^{7}\right) \left(2 \times 10^{−4}\right)$

Multiply and write your answer in decimal form.

$\frac{6 \times 10^{9}}{2 \times 10^{−1}}$

Find the greatest common factor.

$5 n , 45$

Find the greatest common factor.

$12 x^{2} , 20 x^{3} , 36 x^{4}$

Factor the greatest common factor from each polynomial.

$16 u - 24$

Factor the greatest common factor from each polynomial.

$6 p^{2} + 6 p$

Factor the greatest common factor from each polynomial.

$−9 a^{5} - 9 a^{3}$

Factor the greatest common factor from each polynomial.

$5 y^{2} - 55 y + 45$

For the polynomial $8 y^{4} - 3 y^{2} + 1$

1. ⓐ Is it a monomial, binomial, or trinomial?
2. ⓑ What is its degree?

Simplify each expression.

$\left(10 x^{2} - 3 x + 5\right) - \left(4 x^{2} - 6\right)$

Simplify each expression.

$n \cdot n^{4}$

Simplify each expression.

$\left(8 x y^{3}\right) \left(−6 x^{4} y^{6}\right)$

Simplify each expression.

$\left(s + 8\right) \left(s + 9\right)$

Simplify each expression.

$\left(11 a - 6\right) \left(5 a - 1\right)$

Simplify each expression.

$\left(4 a + 9 b\right) \left(6 a - 5 b\right)$

Simplify each expression.

$\left(\frac{x^{3} \cdot x^{9}}{x^{5}}\right)^{2}$

Simplify each expression.

$\frac{24 r^{3} s}{ 6 r^{2} s^{7}}$

Simplify each expression.

$\left(15 x y^{3} - 35 x^{2} y\right) \div 5 x y$

Simplify each expression.

$\left(2 y\right)^{−3}$

Simplify each expression.

$\frac{x^{4}}{ x^{−5}}$

Factor the greatest common factor from each polynomial.

$−6 x^{2} - 30 x$

Factor the greatest common factor from each polynomial.

Convert $5.25 \times 10^{−4}$ to decimal form.

Simplify, and write your answer in decimal form.

$\frac{9 \times 10^{4}}{3 \times 10^{−1}}$