# 1.4 Polynomials  (Page 4/15)

 Page 4 / 15
$\begin{array}{ccc}\hfill \left(x+5\right)\left(x-5\right)& =& {x}^{2}-25\hfill \\ \hfill \left(x+11\right)\left(x-11\right)& =& {x}^{2}-121\hfill \\ \hfill \left(2x+3\right)\left(2x-3\right)& =& 4{x}^{2}-9\hfill \end{array}$

Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.

Is there a special form for the sum of squares?

No. The difference of squares occurs because the opposite signs of the binomials cause the middle terms to disappear. There are no two binomials that multiply to equal a sum of squares.

## Difference of squares

When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.

$\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$

Given a binomial multiplied by a binomial with the same terms but the opposite sign, find the difference of squares.

1. Square the first term of the binomials.
2. Square the last term of the binomials.
3. Subtract the square of the last term from the square of the first term.

## Multiplying binomials resulting in a difference of squares

Multiply $\text{\hspace{0.17em}}\left(9x+4\right)\left(9x-4\right).$

Square the first term to get $\text{\hspace{0.17em}}{\left(9x\right)}^{2}=81{x}^{2}.\text{\hspace{0.17em}}$ Square the last term to get $\text{\hspace{0.17em}}{4}^{2}=16.\text{\hspace{0.17em}}$ Subtract the square of the last term from the square of the first term to find the product of $\text{\hspace{0.17em}}81{x}^{2}-16.$

Multiply $\text{\hspace{0.17em}}\left(2x+7\right)\left(2x-7\right).$

$4{x}^{2}-49$

## Performing operations with polynomials of several variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

## Multiplying polynomials containing several variables

Multiply $\text{\hspace{0.17em}}\left(x+4\right)\left(3x-2y+5\right).$

Follow the same steps that we used to multiply polynomials containing only one variable.

Multiply $\left(3x-1\right)\left(2x+7y-9\right).$

$\text{\hspace{0.17em}}6{x}^{2}+21xy-29x-7y+9$

Access these online resources for additional instruction and practice with polynomials.

## Key equations

 perfect square trinomial ${\left(x+a\right)}^{2}=\left(x+a\right)\left(x+a\right)={x}^{2}+2ax+{a}^{2}$ difference of squares $\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$

## Key concepts

• A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term. See [link] .
• We can add and subtract polynomials by combining like terms. See [link] and [link] .
• To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products. See [link] .
• FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials. See [link] .
• Perfect square trinomials and difference of squares are special products. See [link] and [link] .
• Follow the same rules to work with polynomials containing several variables. See [link] .

## Verbal

Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

Many times, multiplying two binomials with two variables results in a trinomial. This is not the case when there is a difference of two squares. Explain why the product in this case is also a binomial.

You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?

Use the distributive property, multiply, combine like terms, and simplify.

State whether the following statement is true and explain why or why not: A trinomial is always a higher degree than a monomial.

## Algebraic

For the following exercises, identify the degree of the polynomial.

$7x-2{x}^{2}+13$

2

$14{m}^{3}+{m}^{2}-16m+8$

$-625{a}^{8}+16{b}^{4}$

8

$200p-30{p}^{2}m+40{m}^{3}$

${x}^{2}+4x+4$

2

$6{y}^{4}-{y}^{5}+3y-4$

For the following exercises, find the sum or difference.

$\left(12{x}^{2}+3x\right)-\left(8{x}^{2}-19\right)$

$4{x}^{2}+3x+19$

$\left(4{z}^{3}+8{z}^{2}-z\right)+\left(-2{z}^{2}+z+6\right)$

$\left(6{w}^{2}+24w+24\right)-\left(3w{}^{2}-6w+3\right)$

$3{w}^{2}+30w+21$

$\left(7{a}^{3}+6{a}^{2}-4a-13\right)+\left(-3{a}^{3}-4{a}^{2}+6a+17\right)$

$\left(11{b}^{4}-6{b}^{3}+18{b}^{2}-4b+8\right)-\left(3{b}^{3}+6{b}^{2}+3b\right)$

$11{b}^{4}-9{b}^{3}+12{b}^{2}-7b+8$

$\left(49{p}^{2}-25\right)+\left(16{p}^{4}-32{p}^{2}+16\right)$

For the following exercises, find the product.

$\left(4x+2\right)\left(6x-4\right)$

$24{x}^{2}-4x-8$

$\left(14{c}^{2}+4c\right)\left(2{c}^{2}-3c\right)$

$\left(6{b}^{2}-6\right)\left(4{b}^{2}-4\right)$

$24{b}^{4}-48{b}^{2}+24$

$\left(3d-5\right)\left(2d+9\right)$

$\left(9v-11\right)\left(11v-9\right)$

$99{v}^{2}-202v+99$

$\left(4{t}^{2}+7t\right)\left(-3{t}^{2}+4\right)$

$\left(8n-4\right)\left({n}^{2}+9\right)$

$8{n}^{3}-4{n}^{2}+72n-36$

For the following exercises, expand the binomial.

${\left(4x+5\right)}^{2}$

${\left(3y-7\right)}^{2}$

$9{y}^{2}-42y+49$

${\left(12-4x\right)}^{2}$

${\left(4p+9\right)}^{2}$

$16{p}^{2}+72p+81$

${\left(2m-3\right)}^{2}$

${\left(3y-6\right)}^{2}$

$9{y}^{2}-36y+36$

${\left(9b+1\right)}^{2}$

For the following exercises, multiply the binomials.

$\left(4c+1\right)\left(4c-1\right)$

$16{c}^{2}-1$

$\left(9a-4\right)\left(9a+4\right)$

$\left(15n-6\right)\left(15n+6\right)$

$225{n}^{2}-36$

$\left(25b+2\right)\left(25b-2\right)$

$\left(4+4m\right)\left(4-4m\right)$

$-16{m}^{2}+16$

$\left(14p+7\right)\left(14p-7\right)$

$\left(11q-10\right)\left(11q+10\right)$

$121{q}^{2}-100$

For the following exercises, multiply the polynomials.

$\left(2{x}^{2}+2x+1\right)\left(4x-1\right)$

$\left(4{t}^{2}+t-7\right)\left(4{t}^{2}-1\right)$

$16{t}^{4}+4{t}^{3}-32{t}^{2}-t+7$

$\left(x-1\right)\left({x}^{2}-2x+1\right)$

$\left(y-2\right)\left({y}^{2}-4y-9\right)$

${y}^{3}-6{y}^{2}-y+18$

$\left(6k-5\right)\left(6{k}^{2}+5k-1\right)$

$\left(3{p}^{2}+2p-10\right)\left(p-1\right)$

$3{p}^{3}-{p}^{2}-12p+10$

$\left(4m-13\right)\left(2{m}^{2}-7m+9\right)$

$\left(a+b\right)\left(a-b\right)$

${a}^{2}-{b}^{2}$

$\left(4x-6y\right)\left(6x-4y\right)$

${\left(4t-5u\right)}^{2}$

$16{t}^{2}-40tu+25{u}^{2}$

$\left(9m+4n-1\right)\left(2m+8\right)$

$\left(4t-x\right)\left(t-x+1\right)$

$4{t}^{2}+{x}^{2}+4t-5tx-x$

$\left({b}^{2}-1\right)\left({a}^{2}+2ab+{b}^{2}\right)$

$\left(4r-d\right)\left(6r+7d\right)$

$24{r}^{2}+22rd-7{d}^{2}$

$\left(x+y\right)\left({x}^{2}-xy+{y}^{2}\right)$

## Real-world applications

A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: $\text{\hspace{0.17em}}\left(4x+1\right)\left(8x-3\right)\text{\hspace{0.17em}}$ where x is measured in meters. Multiply the binomials to find the area of the plot in standard form.

$32{x}^{2}-4x-3\text{\hspace{0.17em}}$ m 2

A prospective buyer wants to know how much grain a specific silo can hold. The area of the floor of the silo is $\text{\hspace{0.17em}}{\left(2x+9\right)}^{2}.\text{\hspace{0.17em}}$ The height of the silo is $\text{\hspace{0.17em}}10x+10,$ where x is measured in feet. Expand the square and multiply by the height to find the expression that shows how much grain the silo can hold.

## Extensions

For the following exercises, perform the given operations.

${\left(4t-7\right)}^{2}\left(2t+1\right)-\left(4{t}^{2}+2t+11\right)$

$32{t}^{3}-100{t}^{2}+40t+38$

$\left(3b+6\right)\left(3b-6\right)\left(9{b}^{2}-36\right)$

$\left({a}^{2}+4ac+4{c}^{2}\right)\left({a}^{2}-4{c}^{2}\right)$

${a}^{4}+4{a}^{3}c-16a{c}^{3}-16{c}^{4}$

#### Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin

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