10.6 Introduction to factoring polynomials (Page 2/6)

<< Chapter < Page

Prealgebra

Page 2 / 6

Chapter >> Page >

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

8 x ²

Got questions? Get instant answers now!

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

9 y ³

Got questions? Get instant answers now!

Find the greatest common factor of $14 x^{3}, 8 x^{2}, 10 x .$

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column. Bring down the common factors. Multiply the factors.
	$The GCF of 14 x^{3} and 8 x^{2}, and 10 x is 2 x$

Got questions? Get instant answers now!

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

3 x

Got questions? Get instant answers now!

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

5 m ²

Got questions? Get instant answers now!

Factor the greatest common factor from a polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, $12$ as $2 \cdot 6 or 3 \cdot 4),$ in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

\begin{matrix} 2 (x & + & 7) factors \\ 2 \cdot x & + & 2 \cdot 7 \\ 2 x & + & 14 product \end{matrix}

Here, we will start with a product, like $2 x + 14,$ and end with its factors, $2 (x + 7) .$ To do this we apply the Distributive Property “in reverse”.

Distributive property

If $a, b, c$ are real numbers, then

a (b + c) = a b + a c and a b + a c = a (b + c)

The form on the left is used to multiply. The form on the right is used to factor.

So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!

Factor: $2 x + 14 .$

Solution

Step 1: Find the GCF of all the terms of the polynomial.	Find the GCF of 2x and 14.
Step 2: Rewrite each term as a product using the GCF.	Rewrite 2x and 14 as products of their GCF, 2. $2 x = 2 \cdot x$ $14 = 2 \cdot 7$
Step 3: Use the Distributive Property 'in reverse' to factor the expression.		$2 (x + 7)$
Step 4: Check by multiplying the factors.		Check:

Got questions? Get instant answers now!

Factor: $4 x + 12 .$

4( x + 3)

Got questions? Get instant answers now!

Factor: $6 a + 24 .$

6( a + 4)

Got questions? Get instant answers now!

Notice that in [link] , we used the word factor as both a noun and a verb:

\begin{matrix} Noun & 7 is a factor of 14 \\ Verb & factor 2 from 2 x + 14 \end{matrix}

Factor the greatest common factor from a polynomial.

Find the GCF of all the terms of the polynomial.
Rewrite each term as a product using the GCF.
Use the Distributive Property ‘in reverse’ to factor the expression.
Check by multiplying the factors.

Factor: $3 a + 3 .$

Solution



Rewrite each term as a product using the GCF.
Use the Distributive Property 'in reverse' to factor the GCF.
Check by multiplying the factors to get the original polynomial.

Got questions? Get instant answers now!

Factor: $9 a + 9 .$

9( a + 1)

Got questions? Get instant answers now!

Factor: $11 x + 11 .$

11( x + 1)

Got questions? Get instant answers now!

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Factor: $12 x - 60 .$

Solution



Rewrite each term as a product using the GCF.
Factor the GCF.
Check by multiplying the factors.

Got questions? Get instant answers now!

Factor: $11 x - 44 .$

11( x − 4)

Got questions? Get instant answers now!

Factor: $13 y - 52 .$

13( y − 4)

Got questions? Get instant answers now!

Now we’ll factor the greatest common factor from a trinomial . We start by finding the GCF of all three terms.

Factor: $3 y^{2} + 6 y + 9 .$

Solution



Rewrite each term as a product using the GCF.
Factor the GCF.
Check by multiplying.

Got questions? Get instant answers now!

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

4( y ² + 2 y + 3)

Got questions? Get instant answers now!

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

6( x ² + 7 x − 2)

Got questions? Get instant answers now!

In the next example, we factor a variable from a binomial .

Factor: $6 x^{2} + 5 x .$

Solution

	$6 x^{2} + 5 x$
Rewrite each term as a product.
Factor the GCF.	$x (6 x + 5)$
Check by multiplying.
$x (6 x + 5)$ $x \cdot 6 x + x \cdot 5$ $6 x^{2} + 5 x ✓$

Got questions? Get instant answers now!

Questions & Answers

differentiate between demand and supply giving examples

Lambiv Reply

differentiated between demand and supply using examples

Lambiv

what is labour ?

Lambiv

how will I do?

Venny Reply

how is the graph works?I don't fully understand

Rezat Reply

information

Eliyee

devaluation

Eliyee

WARKISA

hi guys good evening to all

Lambiv

multiple choice question

Aster Reply

appreciation

Eliyee

explain perfect market

Lindiwe Reply

In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,

Ezea

What is ceteris paribus?

Shukri Reply

other things being equal

AI-Robot

When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab

Kelo

Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)

Kelo

yes,thank you

Shukri

Can I ask you other question?

Shukri

what is monopoly mean?

Habtamu Reply

What is different between quantity demand and demand?

Shukri Reply

Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded

Ezea

Shukri

how do you save a country economic situation when it's falling apart

Lilia Reply

what is the difference between economic growth and development

Fiker Reply

Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.

Shukri

production function means

Jabir

What do you think is more important to focus on when considering inequality ?

Abdisa Reply

any question about economics?

Awais Reply

sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏

Asui

it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs

Awais

thank you so much 👍 sir

Asui

In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p

Cornelius

Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .

Feyisa Reply

Answer

Feyisa

Jabir

the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema

Gsbwnw Reply

suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product

Abdureman

types of unemployment

Yomi Reply

What is the difference between perfect competition and monopolistic competition?

Mohammed

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Prealgebra' conversation and receive update notifications?

Ask

©flickr:	Math for Economists MCQ By Tony Pizur Start Quiz
	41 Biology 41 Osmotic Regulation and Excretion MCQ By OpenStax Start Quiz
	1 Microeconomics 01 What Is Economics? By OpenStax Start Flashcards
	Introductory statistics By OpenStax Read Online Course
	9 Lec:9 Descriptive Statistics By Janet Forrester Start Quiz
©flickr: BK	Self Confidence By Miranda Reising Start Quiz
©flickr: Francisco	U.S. Civil War Pre-test By Danielle Stephens Start Quiz
©flickr: quinn	Neuroanatomy By George Turner Start Quiz
©flickr:	Biology 1 By Jill Zerressen Start Quiz
	1 Neuroscience Exam 2004 1 By David Corey Start Exam