1.5 Factoring polynomials (Page 2/6)

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

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

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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 $(x - 3) (x + 5) .$

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

$(x −6) (x −1)$

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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}

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Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

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what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

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Why qualify 28 supplies

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what are explicit costs

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out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

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economic overview notes

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identify a demand and a supply curve

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i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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