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### Course: Digital SAT Math > Unit 12

Lesson 3: Operations with polynomials: advanced# Operations with polynomials | Lesson

A guide to operations with polynomials on the digital SAT

## What are polynomial expressions?

A polynomial expression has one or more terms with a coefficient, a variable base, and an exponent.

is a binomial. The exponent of the term${3}{{x}}^{{4}}{+2}{x}$ is${2}{x}$ (${1}$ ).${x}={{x}}^{{1}}$ is a trinomial.${3}{{x}}^{{4}}{+2}{x}{+7}$ is a constant term. We can also think of${7}$ as an exponential term with an exponent of${7}$ . Since${0}$ ,${{x}^{{0}}}=1$ is equivalent to${7}$ .${7}{{x}}^{{0}}$

In this lesson, we'll learn to add, subtract, and multiply polynomials.

**You can learn anything. Let's do this!**

## How do I add and subtract polynomials?

### Adding polynomials

### What should I be careful of when adding and subtracting polynomials?

While we can add and subtract any polynomials, we can only combine

**like terms**, which must have:- The same variable base
- The same exponent

For example, we can combine the terms ${2}{{x}}^{{3}}$ and ${4}{{x}}^{{3}}$ because they have the same variable base, ${x}$ , and the same exponent, ${3}$ . However, we ${2}{{x}}^{{2}}$ and ${2}{{x}}^{{3}}$ because they have different exponents, ${2}$ and ${3}$ .

*cannot*combine the termsWhen we combine like terms, only the coefficients change. Both the base and the exponent remain the same. For example, when adding ${2}{{x}}^{{3}}$ and ${4}{{x}}^{{3}}$ , the ${{x}}^{{3}}$ part of the terms remain the same, and we add only ${2}$ and ${4}$ when combining the terms:

When subtracting polynomials, make sure to distribute the negative sign as needed. For example, when subtracting the polynomial $-2{x}^{2}-7$ , the negative sign from the subtraction is distributed to both $-2{x}^{2}$ and $-7$ , which means:

Subtracting $-2{x}^{2}-7$ is equivalent to adding $2{x}^{2}+7$ !

To add or subtract two polynomials:

- Group like terms.
- For each group of like terms, add or subtract the coefficients while keeping both the base and the exponent the same.
- Write the combined terms in order of decreasing power.

### Try it!

## How do I multiply polynomials?

### Multiplying binomials

### What should I be careful of when multiplying polynomials?

When multiplying two polynomials, we must make sure to distribute each term of one polynomial to all the terms of the other polynomial. For example:

The total number of products we need to calculate is equal to the product of the number of terms in each polynomial. Multiplying two binomials requires $2\cdot 2=4$ products, as shown above. Multiplying a monomial and a trinomial requires $1\cdot 3=3$ products; multiplying a binomial and a trinomial requires $2\cdot 3=6$ products.

When multiplying two binomials, we can also use the
mnemonic FOIL to account for all four multiplications. For $({ax}+{b})({cx}+{d})$ :

- Multiply the
**First**terms ( )${ax}\cdot {cx}$ - Multiply the
**Outer**terms ( )${ax}\cdot {d}$ - Multiply the
**Inner**terms ( )${b}\cdot {cx}$ - Multiply the
**Last**terms ( )${b}\cdot {d}$

When multiplying terms of polynomial expressions with the same base:

- Multiply the coefficients, or multiply the coefficient and the constant.
- Keep the base the same.
- Add the exponents.

To multiply two polynomials:

- Distribute the terms.
- Multiply the distributed terms according to the exponent rules above.
- Group like terms.
- For each group of like terms, add or subtract the coefficients while keeping both the base and the exponent the same.
- Write the combined terms in order of decreasing power.

#### Let's look at some examples!

What is the product of $2x-1$ and $x-5$ ?

What is the product of $3x$ and ${x}^{2}-4x+9$ ?

### Try it!

## Your turn!

## Things to remember

The mnemonic FOIL for multiplying two binomials:

- Multiply the
**First**terms - Multiply the
**Outer**terms - Multiply the
**Inner**terms - Multiply the
**Last**terms

## Want to join the conversation?

- polynomials are a breath of fresh air after radicals and rationals(90 votes)
- You're dang right!(10 votes)

- I believe the previous lesson was way harder, might wanna reconsider the structure a bit, so that each lesson would help you understand the next one better. great work anyway(91 votes)
- 7 days remaining to my SAT(29 votes)
- Same! Wish me luck 😭(7 votes)

- I have my SAT exam in 9 hours. Good luck to me.(25 votes)
- I hope it went well.(11 votes)

- This was so easy I was singing Hamilton while doing this. May all math questions be likewise!(23 votes)
- After the last few lessons, it's great to not be confused for once.(18 votes)
- i am in love with KHAN Academey(13 votes)
- but i am in love with sal,axaxaxa(11 votes)

- the less commented topic in Khan Academy(11 votes)
- Extremely fantastic lessons . Thank you Khan Academy for that(8 votes)
- does anyone do the box method for polynomials? i was taught that and doing this other method is hard(6 votes)
- Yes. The box method is good as well it's all about what you feel comfortable using.(3 votes)