# 1.4 Polynomials  (Page 4/15)

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

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] .
• 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.

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

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

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

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

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

given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
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jai
which language is that
Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
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SORIE
I speak French
Abdel
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SORIE
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls