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Course: Digital SAT Math > Unit 8
Lesson 3: Operations with polynomials: mediumOperations 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 is ( ). is a trinomial. is a constant term. We can also think of as an exponential term with an exponent of . Since , is equivalent to .
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 and because they have the same variable base, , and the same exponent, . However, we cannot combine the terms and because they have different exponents, and .
When we combine like terms, only the coefficients change. Both the base and the exponent remain the same. For example, when adding and , the part of the terms remain the same, and we add only and when combining the terms:
When subtracting polynomials, make sure to distribute the negative sign as needed. For example, when subtracting the polynomial , the negative sign from the subtraction is distributed to both and , which means:
Subtracting is equivalent to adding !
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 products, as shown above. Multiplying a monomial and a trinomial requires products; multiplying a binomial and a trinomial requires products.
When multiplying two binomials, we can also use the
mnemonic FOIL to account for all four multiplications. For :
- Multiply the First terms (
) - Multiply the Outer terms (
) - Multiply the Inner terms (
) - Multiply the Last terms (
)
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 and ?
What is the product of and ?
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?
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- This is a section for questions(105 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(58 votes)
- polynomials are a breath of fresh air after radicals and rationals(33 votes)
- 7 days remaining to my SAT(22 votes)
- Same! Wish me luck 😭(3 votes)
- I have my SAT exam in 9 hours. Good luck to me.(13 votes)
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- After the last few lessons, it's great to not be confused for once.(7 votes)
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