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### Course: Digital SAT Math>Unit 4

Lesson 3: Operations with polynomials: foundations

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

### 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 cannot combine the terms $2{x}^{2}$ and $2{x}^{3}$ because they have different exponents, $2$ and $3$.
When 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:
$\begin{array}{rl}2{x}^{3}+4{x}^{3}& =\left(2+4\right){x}^{3}\\ \\ & =6{x}^{3}\end{array}$
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:
$\begin{array}{rl}& 5{x}^{2}-\left(-2{x}^{2}-7\right)\\ \\ & =5{x}^{2}+\left(-1\right)\left(-2{x}^{2}\right)+\left(-1\right)\left(-7\right)\\ \\ & =5{x}^{2}+2{x}^{2}+7\\ \\ & =\left(5+2\right){x}^{2}+7\\ \\ & =7{x}^{2}+7\end{array}$
Subtracting $-2{x}^{2}-7$ is equivalent to adding $2{x}^{2}+7$!
To add or subtract two polynomials:
1. Group like terms.
2. For each group of like terms, add or subtract the coefficients while keeping both the base and the exponent the same.
3. Write the combined terms in order of decreasing power.

### Try it!

TRY: Match the equivalent expressions
Because $9{x}^{2}$ and $3{x}^{2}$ have
and
, the two terms
into a single term.
Because $4{y}^{4}$ and $4y$ have
, the two terms
into a single term.

TRY: Match the equivalent expressions
Match each polynomial expression below with an equivalent expression.

## How do I multiply polynomials?

### Multiplying binomials

Multiplying binomialsSee video transcript

### 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:
$\begin{array}{rl}& \left(ax+b\right)\left(cx+d\right)\\ \\ =& \phantom{\rule{0.167em}{0ex}}\left(ax\right)\left(cx+d\right)+\left(b\right)\left(cx+d\right)\\ \\ =& \phantom{\rule{0.167em}{0ex}}\left(ax\right)\left(cx\right)+\left(ax\right)\left(d\right)+\left(b\right)\left(cx\right)+\left(b\right)\left(d\right)\end{array}$
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 $\left(ax+b\right)\left(cx+d\right)$:
1. Multiply the First terms ($ax\cdot cx$)
2. Multiply the Outer terms ($ax\cdot d$)
3. Multiply the Inner terms ($b\cdot cx$)
4. Multiply the Last terms ($b\cdot d$)
When multiplying terms of polynomial expressions with the same base:
1. Multiply the coefficients, or multiply the coefficient and the constant.
2. Keep the base the same.
$\begin{array}{rl}a{x}^{m}\cdot b{x}^{n}& =ab\cdot {x}^{m+n}\\ \\ a\cdot b{x}^{n}& =ab\cdot {x}^{n}\end{array}$
To multiply two polynomials:
1. Distribute the terms.
2. Multiply the distributed terms according to the exponent rules above.
3. Group like terms.
4. For each group of like terms, add or subtract the coefficients while keeping both the base and the exponent the same.
5. 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!

TRY: multiply two terms
When multiplying $3x$ and $2{x}^{2}$, we
the coefficients of the terms and
the exponents of $x$.
$\left(3x\right)\left(2{x}^{2}\right)=\phantom{\rule{0.167em}{0ex}}$

TRY: Multiply two binomials using FOIL
Use the table below to FOIL $\left(8x-3\right)\left({x}^{2}+1\right)$.
TermExpressionProduct
First$8x\cdot {x}^{2}$
Outer
$8x$
Inner$-3\cdot {x}^{2}$
Last
$-3$

Which of the following is the sum of ${x}^{2}+5$ and $2{x}^{2}+4x$ ?

Practice: subtract two polynomials to find a coefficient
$\left(9{x}^{2}+5x-1\right)-\left(6{x}^{2}-4x\right)=a{x}^{2}+bx+c$
The equation above is true for all $x$, where $a$, $b$ and $c$ are constants. What is the value of $b$ ?

Practice: multiply two binomials
Which of the following is equivalent to $\left(x+3\right)\left(2x-5\right)$ ?

Practice: multiply two binomials with symbolic coefficients
$\left(ax+3\right)\left(bx+2\right)=9{x}^{2}+21x+6$
In the equation above, $a$ and $b$ are constants. What is the value of $ab$ ?

## Things to remember

The mnemonic FOIL for multiplying two binomials:
1. Multiply the First terms
2. Multiply the Outer terms
3. Multiply the Inner terms
4. Multiply the Last terms
$\begin{array}{rl}a{x}^{m}\cdot b{x}^{n}& =ab\cdot {x}^{m+n}\\ \\ a\cdot b{x}^{n}& =ab\cdot {x}^{n}\end{array}$

## Want to join the conversation?

• freaking easyyy ////
• This is a section for questions
• 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
• polynomials are a breath of fresh air after radicals and rationals
• You're dang right!
• 7 days remaining to my SAT
• I have my SAT exam in 9 hours. Good luck to me.
• I hope it went well.