# 1.4 Polynomials  (Page 2/15)

 Page 2 / 15

## Identifying the degree and leading coefficient of a polynomial

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

1. $3+2{x}^{2}-4{x}^{3}$
2. $5{t}^{5}-2{t}^{3}+7t$
3. $6p-{p}^{3}-2$
1. The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, $\text{\hspace{0.17em}}-4{x}^{3}.\text{\hspace{0.17em}}$ The leading coefficient is the coefficient of that term, $\text{\hspace{0.17em}}-4.$
2. The highest power of t is $\text{\hspace{0.17em}}5,$ so the degree is $\text{\hspace{0.17em}}5.\text{\hspace{0.17em}}$ The leading term is the term containing that degree, $\text{\hspace{0.17em}}5{t}^{5}.\text{\hspace{0.17em}}$ The leading coefficient is the coefficient of that term, $\text{\hspace{0.17em}}5.$
3. The highest power of p is $\text{\hspace{0.17em}}3,$ so the degree is $\text{\hspace{0.17em}}3.\text{\hspace{0.17em}}$ The leading term is the term containing that degree, $\text{\hspace{0.17em}}-{p}^{3},$ The leading coefficient is the coefficient of that term, $\text{\hspace{0.17em}}-1.$

Identify the degree, leading term, and leading coefficient of the polynomial $\text{\hspace{0.17em}}4{x}^{2}-{x}^{6}+2x-6.$

The degree is 6, the leading term is $\text{\hspace{0.17em}}-{x}^{6},$ and the leading coefficient is $\text{\hspace{0.17em}}-1.$

## Adding and subtracting polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, $\text{\hspace{0.17em}}5{x}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}-2{x}^{2}\text{\hspace{0.17em}}$ are like terms, and can be added to get $\text{\hspace{0.17em}}3{x}^{2},$ but $\text{\hspace{0.17em}}3x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}3{x}^{2}\text{\hspace{0.17em}}$ are not like terms, and therefore cannot be added.

Given multiple polynomials, add or subtract them to simplify the expressions.

1. Combine like terms.
2. Simplify and write in standard form.

## Adding polynomials

Find the sum.

$\left(12{x}^{2}+9x-21\right)+\left(4{x}^{3}+8{x}^{2}-5x+20\right)$

Find the sum.

$\left(2{x}^{3}+5{x}^{2}-x+1\right)+\left(2{x}^{2}-3x-4\right)$

$2{x}^{3}+7{x}^{2}-4x-3$

## Subtracting polynomials

Find the difference.

$\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)$

Find the difference.

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

$-11{x}^{3}-{x}^{2}+7x-9$

## Multiplying polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

## Multiplying polynomials using the distributive property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ in $\text{\hspace{0.17em}}2\left(x+7\right)\text{\hspace{0.17em}}$ to obtain the equivalent expression $\text{\hspace{0.17em}}2x+14.\text{\hspace{0.17em}}$ When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

Given the multiplication of two polynomials, use the distributive property to simplify the expression.

1. Multiply each term of the first polynomial by each term of the second.
2. Combine like terms.
3. Simplify.

## Multiplying polynomials using the distributive property

Find the product.

$\left(2x+1\right)\left(3{x}^{2}-x+4\right)$

#### Questions & Answers

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications? By By     By  By   By