1.5 Factoring polynomials (Page 3/6)

College algebra

Page 3 / 6

Factor a. $2 x^{2} + 9 x + 9$ b. $6 x^{2} + x - 1$

a. $(2 x + 3) (x + 3)$ b. $(3 x −1) (2 x + 1)$

Factoring a perfect square trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.

\begin{matrix} a^{2} + 2 a b + b^{2} & = & {(a + b)}^{2} \\ and \\ a^{2} - 2 a b + b^{2} & = & {(a - b)}^{2} \end{matrix}

We can use this equation to factor any perfect square trinomial.

A General Note

Perfect square trinomials

A perfect square trinomial can be written as the square of a binomial:

a^{2} + 2 a b + b^{2} = {(a + b)}^{2}

How To

Given a perfect square trinomial, factor it into the square of a binomial.

Confirm that the first and last term are perfect squares.
Confirm that the middle term is twice the product of $a b .$
Write the factored form as ${(a + b)}^{2} .$

Factoring a perfect square trinomial

Factor $25 x^{2} + 20 x + 4.$

Notice that $25 x^{2}$ and $4$ are perfect squares because $25 x^{2} = {(5 x)}^{2}$ and $4 = 2^{2} .$ Then check to see if the middle term is twice the product of $5 x$ and $2.$ The middle term is, indeed, twice the product: $2 (5 x) (2) = 20 x .$ Therefore, the trinomial is a perfect square trinomial and can be written as ${(5 x + 2)}^{2} .$

Got questions? Get instant answers now!

Try It

Factor $49 x^{2} - 14 x + 1.$

${(7 x −1)}^{2}$

Got questions? Get instant answers now!

Factoring a difference of squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.

a^{2} - b^{2} = (a + b) (a - b)

We can use this equation to factor any differences of squares.

A General Note

Differences of squares

A difference of squares can be rewritten as two factors containing the same terms but opposite signs.

a^{2} - b^{2} = (a + b) (a - b)

How To

Given a difference of squares, factor it into binomials.

Confirm that the first and last term are perfect squares.
Write the factored form as $(a + b) (a - b) .$

Factoring a difference of squares

Factor $9 x^{2} - 25.$

Notice that $9 x^{2}$ and $25$ are perfect squares because $9 x^{2} = {(3 x)}^{2}$ and $25 = 5^{2} .$ The polynomial represents a difference of squares and can be rewritten as $(3 x + 5) (3 x - 5) .$

Got questions? Get instant answers now!

Try It

Factor $81 y^{2} - 100.$

$(9 y + 10) (9 y - 10)$

Got questions? Get instant answers now!

Q&A

Is there a formula to factor the sum of squares?

No. A sum of squares cannot be factored.

Factoring the sum and difference of cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

Similarly, the sum of cubes can be factored into a binomial and a trinomial, but with different signs.

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: S ame O pposite A lways P ositive. For example, consider the following example.

x^{3} - 2^{3} = (x - 2) (x^{2} + 2 x + 4)

The sign of the first 2 is the same as the sign between $x^{3} - 2^{3} .$ The sign of the $2 x$ term is opposite the sign between $x^{3} - 2^{3} .$ And the sign of the last term, 4, is always positive .

A General Note

Sum and difference of cubes

We can factor the sum of two cubes as

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

We can factor the difference of two cubes as

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

Questions & Answers

what does preconceived mean

sammie Reply

physiological Psychology

Nwosu Reply

How can I develope my cognitive domain

Amanyire Reply

why is communication effective

Dakolo Reply

Communication is effective because it allows individuals to share ideas, thoughts, and information with others.

effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.

it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.

miss

Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment

Wekolamo Reply

please i need answer

Wekolamo

because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).

Manix Reply

Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!

ARC

A child is a member of community not society elucidate ?

JESSY Reply

Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?

Simon Reply

compare and contrast skinner's perspective on personality development on freud

namakula Reply

Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning

war

explain how nature and nurture affect the development and later the productivity of an individual.

Amesalu Reply

nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.

Samuel

I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills

Zyryn Reply

good👍

Jonathan

and having a good philosophy of the world is like a sandwich and a peanut butter 👍

Jonathan

generally amnesi how long yrs memory loss

Kelu Reply

interpersonal relationships

Abdulfatai Reply

What would be the best educational aid(s) for gifted kids/savants?

Heidi Reply

treat them normal, if they want help then give them. that will make everyone happy

Saurabh

What are the treatment for autism?

Magret Reply

hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these

Jharna

continue.. sometimes due to these physical problems..the diagnosis may be misdiagnosed. treatment for autism. well it depends on the severity. since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child

Jharna

child interact with other kids under doc supervision. play therapy. speech therapy. Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game. string and beads game. the more you interact with the child the more effective

Jharna

results you'll get.. please consult a therapist to know what suits best on your child. and last as a parent. I know sometimes it's overwhelming to guide a special kid. but trust the process and be strong and patient as a parent.

Jharna

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

Learn

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

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

Notification Switch

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

Ask

	6 Factoring the sum and difference of cubes By OpenStax Read Online Course
	4 Factoring a perfect square trinomial By OpenStax Read Online Course
	12 Dr Dowers Endocrinology By Brooke Delaney Start Exam
	1 Neuroanatomy 01 The Cranial Nerves By Stephen Voron Start Quiz
	1 Timeshare 1 By Jams Kalo Start Quiz
	11 Biology 11 Meiosis Sexual Reproduction MCQ By OpenStax Start Quiz
©flickr: Hey	Anatomy Physiology Final By Joli Julianna Start Test
	2 Week 2 Social Psych By Yacoub Jayoghli Start Quiz
	1 Microeconomics 01 What Is Economics? By OpenStax Start Flashcards
	9 Domain Driven Design By JavaChamp Team Start Quiz
	Lean Startup Quiz By Yasser Ibrahim Start Quiz
	SCJP Online Exam 310-065 By Prateek Ashtikar Start Quiz