1.5 Factoring polynomials (Page 3/6)

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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 ${(5 x + 2)}^{2} .$

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

${(7 x −1)}^{2}$

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

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

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

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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

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

Petrus Reply

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

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

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

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

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

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