1.5 Factoring polynomials (Page 2/6)

<< Chapter < Page

Algebra and trigonometry

Page 2 / 6

Chapter >> Page >

Try It

Factor $x (b^{2} - a) + 6 (b^{2} - a)$ by pulling out the GCF.

$(b^{2} - a) (x + 6)$

Got questions? Get instant answers now!

Factoring a trinomial with leading coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial $x^{2} + 5 x + 6$ has a GCF of 1, but it can be written as the product of the factors $(x + 2)$ and $(x + 3) .$

Trinomials of the form $x^{2} + b x + c$ can be factored by finding two numbers with a product of $c$ and a sum of $b .$ The trinomial $x^{2} + 10 x + 16,$ for example, can be factored using the numbers $2$ and $8$ because the product of those numbers is $16$ and their sum is $10.$ The trinomial can be rewritten as the product of $(x + 2)$ and $(x + 8) .$

A General Note

Factoring a trinomial with leading coefficient 1

A trinomial of the form $x^{2} + b x + c$ can be written in factored form as $(x + p) (x + q)$ where $p q = c$ and $p + q = b .$

Q&A

Can every trinomial be factored as a product of binomials?

No. Some polynomials cannot be factored. These polynomials are said to be prime.

How To

Given a trinomial in the form $x^{2} + b x + c,$ factor it.

List factors of $c .$
Find $p$ and $q,$ a pair of factors of $c$ with a sum of $b .$
Write the factored expression $(x + p) (x + q) .$

Factoring a trinomial with leading coefficient 1

Factor $x^{2} + 2 x - 15.$

We have a trinomial with leading coefficient $1, b = 2,$ and $c = −15.$ We need to find two numbers with a product of $−15$ and a sum of $2.$ In [link] , we list factors until we find a pair with the desired sum.


Factors of $−15$	Sum of Factors
$1, −15$	$−14$
$−1, 15$	14
$3, −5$	$−2$
$−3, 5$	2

Now that we have identified $p$ and $q$ as $−3$ and $5,$ write the factored form as $(x - 3) (x + 5) .$

Got questions? Get instant answers now!

Q&A

Does the order of the factors matter?

No. Multiplication is commutative, so the order of the factors does not matter.

Try It

Factor $x^{2} - 7 x + 6.$

$(x −6) (x −1)$

Got questions? Get instant answers now!

Factoring by grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial $2 x^{2} + 5 x + 3$ can be rewritten as $(2 x + 3) (x + 1)$ using this process. We begin by rewriting the original expression as $2 x^{2} + 2 x + 3 x + 3$ and then factor each portion of the expression to obtain $2 x (x + 1) + 3 (x + 1) .$ We then pull out the GCF of $(x + 1)$ to find the factored expression.

A General Note

Factor by grouping

To factor a trinomial in the form $a x^{2} + b x + c$ by grouping, we find two numbers with a product of $a c$ and a sum of $b .$ We use these numbers to divide the $x$ term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

How To Given a trinomial in the form $a x^{2} + b x + c,$ factor by grouping.

List factors of $a c .$
Find $p$ and $q,$ a pair of factors of $a c$ with a sum of $b .$
Rewrite the original expression as $a x^{2} + p x + q x + c .$
Pull out the GCF of $a x^{2} + p x .$
Pull out the GCF of $q x + c .$
Factor out the GCF of the expression.

Factoring a trinomial by grouping

Factor $5 x^{2} + 7 x - 6$ by grouping.

We have a trinomial with $a = 5, b = 7,$ and $c = −6.$ First, determine $a c = −30.$ We need to find two numbers with a product of $−30$ and a sum of $7.$ In [link] , we list factors until we find a pair with the desired sum.


Factors of $−30$	Sum of Factors
$1, −30$	$−29$
$−1, 30$	29
$2, −15$	$−13$
$−2, 15$	13
$3, −10$	$−7$
$−3, 10$	7

So $p = −3$ and $q = 10.$

\begin{matrix} 5 x^{2} - 3 x + 10 x - 6 & Rewrite the original expression as a x^{2} + p x + q x + c . \\ x (5 x - 3) + 2 (5 x - 3) & Factor out the GCF of each part . \\ (5 x - 3) (x + 2) & Factor out the GCF ​ of the expression . \end{matrix}

Got questions? Get instant answers now!

Questions & Answers

it is the relatively stable flow of income

Chidubem Reply

what is circular flow of income

Divine Reply

branches of macroeconomics

SHEDRACK Reply

what is Flexible exchang rate?

poudel Reply

is gdp a reliable measurement of wealth

Atega Reply

introduction to econometrics

Husseini Reply

mostafa

LEMLEM

hello

Sammol

Mahesh

Ruqayat

Hi fellas

Nyawa

hey

Sammol

God

hello

Jahara

Good morning

Jorge

abubakar

Nmesoma

Mahesh

Tom

Why is unemployment rate never zero at full employment?

Priyanka Reply

bcoz of existence of frictional unemployment in our economy.

Umashankar

what is flexible exchang rate?

poudel

due to existence of the pple with disabilities

Abdulraufu

the demand of a good rises, causing the demand for another good to fall

Rushawn Reply

is it possible to leave every good at the same level

Joseph

I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish

Anuolu

is not really possible to let the value of a goods to be same at the same time.....

Salome

Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.

Geetha Reply

Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.

Waeth

discus major problems of macroeconomics

Alii Reply

what is the problem of macroeconomics

Yoal

Economic growth Stable prices and low unemployment

Ephraim

explain inflationcause and itis degre

Miresa Reply

what is inflation

Getu

increase in general price levels

WEETO

Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?

Chisomo Reply

how to calculate actual output?

Chisomo

how to calculate the equilibrium income

Beshir

Criteria for determining money supply

Thapase Reply

who we can define macroeconomics in one line

Muhammad

Aggregate demand

Mohammed

C=k100 +9y and i=k50.calculate the equilibrium level of output

Mercy Reply

Isiaka

Geli

Man

👋

Bahunda

hy how are you?

Man

Amisha

how are you guys

Sekou

f9 guys

Amisha

how are you guys

Sekou

ys am also fine

Amisha

fine and you guys

Geli

from Nepal

Amisha

nawalparasi district from belatari

Amisha

nd u

Amisha

I am Camara from Guinea west Africa... happy to meet you guys here

Sekou

ma management ho

Amisha

ahile becheclor ho

Amisha

hjr ktm bta ho ani k kaam grnu hunxa tw

Amisha

belatari

Amisha

1st year ho

Amisha

nd u

Amisha

ahh

Amisha

kaha biratnagar

Amisha

kina k vo

Amisha

money as unit of account means what?

Kalombe

A unit of account is something that can be used to value goods and services and make calculations

Jim

all of you please speak in English I can't understand you're language

Muhammad

I want to know how can we define macroeconomics in one line

Muhammad

it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500

Kalombe

Mercy is it clear?😋

Kalombe

hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.

Galge Reply

if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️

Kalombe

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

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 2

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, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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

Notification Switch

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

Ask

	3 Factoring by grouping By OpenStax Read Online Course
	1 Factoring the greatest common factor of a polynomial By OpenStax Read Online Course
	English Vocabulary By Jordon Humphreys Start Quiz
	Introduction to Mechanics MCQ By Saylor Foundation Start Quiz
	28 AP 28 Development Inheritance Essay By OpenStax Start Flashcards
	18 AP 18 Cardiovascular System Blood MCQ By OpenStax Start Quiz
	23 AP Key Terms 23 The Digestive System By OpenStax Start Key Terms
	Biology Exam Final By Savannah Parrish Start Exam
	27 AP Key Terms 27 The Reproductive System By OpenStax Start Key Terms
	23 Pesticides/Small animal poison Test By Brooke Delaney Start Exam
	Social Work midterm By Katy Pratt Start Exam
©flickr:	CTE/STEM Technology Literacy Test By Sebastian Sieczko... Start Quiz