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## SAT

### Course: SAT > Unit 6

Lesson 4: Passport to Advanced Math: lessons by skill- Solving quadratic equations | Lesson
- Interpreting nonlinear expressions | Lesson
- Quadratic and exponential word problems | Lesson
- Manipulating quadratic and exponential expressions | Lesson
- Radicals and rational exponents | Lesson
- Radical and rational equations | Lesson
- Operations with rational expressions | Lesson
- Operations with polynomials | Lesson
- Polynomial factors and graphs | Lesson
- Graphing quadratic functions | Lesson
- Graphing exponential functions | Lesson
- Linear and quadratic systems | Lesson
- Structure in expressions | Lesson
- Isolating quantities | Lesson
- Function Notation | Lesson

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# Operations with polynomials | Lesson

## What are polynomial expressions, and how frequently do they appear on the test?

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

start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript is a . We'll also frequently see and .

- start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript, start color #7854ab, plus, 2, end color #7854ab, start color #ca337c, x, end color #ca337c is a binomial. The exponent of the term start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c is start color #208170, 1, end color #208170 (start color #ca337c, x, end color #ca337c, equals, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 1, end color #208170, end superscript).
- start color #7854ab, 3, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 4, end color #208170, end superscript, start color #7854ab, plus, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start color #a75a05, plus, 7, end color #a75a05 is a trinomial. start color #a75a05, 7, end color #a75a05 is a constant term. We can also think of start color #a75a05, 7, end color #a75a05 as an exponential term with an exponent of start color #208170, 0, end color #208170. Since start color #ca337c, x, start superscript, start color #208170, 0, end color #208170, end superscript, end color #ca337c, equals, 1, start color #a75a05, 7, end color #a75a05 is equivalent to start color #7854ab, 7, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 0, end color #208170, end superscript.

On your official SAT, you'll likely see

**1 to 3 questions**that test your ability to add, subtract, and multiply polynomials. In addition, you'll see polynomial expressions as a part of many other Passport to Advanced Math questions.**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 start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript and start color #7854ab, 4, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript because they have the same variable base, start color #ca337c, x, end color #ca337c, and the same exponent, start color #208170, 3, end color #208170. However, we

*cannot*combine the terms start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 2, end color #208170, end superscript and start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript because they have different exponents, start color #208170, 2, end color #208170 and start color #208170, 3, end color #208170.When we combine like terms, only the coefficients change. Both the base and the exponent remain the same. For example, when adding start color #7854ab, 2, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript and start color #7854ab, 4, end color #7854ab, start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript, the start color #ca337c, x, end color #ca337c, start superscript, start color #208170, 3, end color #208170, end superscript part of the terms remain the same, and we add only start color #7854ab, 2, end color #7854ab and start color #7854ab, 4, end color #7854ab when combining the terms:

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

Subtracting minus, 2, x, squared, minus, 7 is equivalent to adding 2, x, squared, plus, 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, dot, 2, equals, 4 products, as shown above. Multiplying a monomial and a trinomial requires 1, dot, 3, equals, 3 products; multiplying a binomial and a trinomial requires 2, dot, 3, equals, 6 products.

When multiplying two binomials, we can also use the
mnemonic FOIL to account for all four multiplications. For left parenthesis, start color #7854ab, a, x, end color #7854ab, plus, start color #ca337c, b, end color #ca337c, right parenthesis, left parenthesis, start color #208170, c, x, end color #208170, plus, start color #a75a05, d, end color #a75a05, right parenthesis:

- Multiply the
**First**terms (start color #7854ab, a, x, end color #7854ab, dot, start color #208170, c, x, end color #208170) - Multiply the
**Outer**terms (start color #7854ab, a, x, end color #7854ab, dot, start color #a75a05, d, end color #a75a05) - Multiply the
**Inner**terms (start color #ca337c, b, end color #ca337c, dot, start color #208170, c, x, end color #208170) - Multiply the
**Last**terms (start color #ca337c, b, end color #ca337c, dot, start color #a75a05, d, end color #a75a05)

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 2, x, minus, 1 and x, minus, 5 ?

What is the product of 3, x and x, squared, minus, 4, x, plus, 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?

- where did you get the 9 from in the last question?(2 votes)
- They first multiplied the (ax + 3) with the (bx + 2) to get ( abx^2 + 3bx + 2ax + 6 ). You don't really have to, but you can then group the middle two terms as (3b + 2a)x to get (abx^2 + (3b + 2a)x + 6. Then you just compare the two equations and notice that the first term of the second equation has an x^2 and that the first term of the equation you solved for has an X^2 as well, you can assume that (ab) must be equal to 9 because (abx^2) is equal to (9x^2). I'm no expert at this, but a lot of these SAT questions require you to be able to look at an equation with multiple unknown values and be able to correlate those with the coefficients of the ones you do know from a given equation (or one that you solved for). An example of this would be if you had a question that states 3x - 5 = ax - 5. In this case we can assume that (a) has to be equal to (3), because then you would get 3x - 5 = 3x - 5. It gets a bit tricky when the values don't match exactly like 3x-5 = ax-10. In that example we can assume that the second equation is just all of the first equation multiplied by 2, so a has to equal 6 because 3x-5 = 6x-10.(10 votes)

- when you are multiplying does it matter what side you multiply first.(1 vote)
- Not really, as long as you follow the FOIL procedure it should be fine, in other words If you do (3x+4)(2x+5) it should be the same as(2x+5)(3x+4)...however im not an expert(4 votes)

- Why don't I understand any of this(4 votes)
- Basically when you are multiplying a monomial you do it normally, but when you are doing with binomial or trinomial it is a little different. Just review the examples in khan academy. It is perfectly explained step by step.(0 votes)

- do we simplify the final result of the polynomials?(2 votes)
- Where can I find the steps to solve

xsquared + ysquared = 100

xy. = 48

I know the answer is 8 and 6, but what are the steps to solve for the answer?

Dave(1 vote)