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

how do you get the 2/50

Abba Reply

number of sport play by 50 student construct discrete data

Aminu Reply

width of the frangebany leaves on how to write a introduction

Theresa Reply

Solve the mean of variance

Veronica Reply

Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.

kenneth

what is error

Yakuba Reply

Is mistake done to something

Vutshila

anas

What is the life teble

anas

Jibrin

statistics is the analyzing of data

Tajudeen Reply

what is statics?

Zelalem Reply

how do you calculate mean

Gloria Reply

diveving the sum if all values

Shaynaynay

let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?

Fisaye Reply

what is statistics

Akhisani Reply

data collected all over the world

Shaynaynay

construct a less than and more than table

Imad Reply

The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.

Aschalew Reply

Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?

Akshay Reply

Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits

Akshay

Bright

Sorry i want to learn more about this question

Bright

Someone help

Bright

a= 0.20233 b=0.3384

Sufiyan

Shaynaynay

How do I interpret level of significance?

Mohd Reply

It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value

Shivam

how skewness and kurtosis are used in statistics

Owen Reply

yes what is it

Taneeya

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

	4 Factoring a perfect square trinomial By OpenStax Read Online Course
	2 Factoring a trinomial with leading coefficient 1 By OpenStax Read Online Course
	26 AP 26 Fluid, Electrolyte, Acid-Base Balance MCQ By OpenStax Start Quiz
	25 AP Key Terms 25 The Urinary System By OpenStax Start Key Terms
	2 SCEA Java Architect By JavaChamp Team Start Exam
	1 SCJP/OCJP Java Certification By JavaChamp Team Start Exam
	11 AP 11 Muscular System Essay By OpenStax Start Flashcards
	NCE Ch 09 Research and Program Evaluation By Anh Dao Start Quiz
	1 Psychology Concept Test By John Gabrieli Start Test
	12 Neuroanatomy 12 The Basal Ganglia By Stephen Voron Start Quiz
	English Composition 2 Final Practice By Madison Christian Start Test
	5 AP 05 Integumentary System MCQ By OpenStax Start Quiz