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## Algebra 2

### Course: Algebra 2 > Unit 1

Lesson 5: Multiplying binomials by polynomials# Multiplying binomials by polynomials review

CCSS.Math: ,

A review of how to multiply binomials like 1 + x by polynomials with more than two terms like x^2 - 5x - 6

We already know how to multiply binomials like left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 7, right parenthesis. In this article, we review a slightly more complicated skill: multiplying binomials by polynomials with more than two terms.

### Example

**Expand and simplify.**

Apply the distributive property.

Apply the distributive property again.

*Notice the pattern. We multiplied each term in the binomial by each term in the trinomial.*

Simplify.

*Want to learn more about multiplying binomials and polynomials? Check out this video.*

*Want more practice? Check out this exercise.*

## Want to join the conversation?

- How do you simplify an equation(9 votes)
- You just need to combine like terms. Example- t to the fourth plus t to the fourth; 2w squared plus w squared. I hope this helps and God bless!(35 votes)

- in the vid they never clearly said which goes first they just said its the only one cubed and do I do y squared first or y alone first(7 votes)
- Generally, you should write your answer in standard form. This would have the term with y^2 first, then y, then the constant. However, it is usually not required unless the instructions specify your answer must be in standard form.(11 votes)

- What do you think is the best way to multiply and simplify a problem?(3 votes)
- In my opinion, the area model is easier because I can see out all the terms and not get things mixed up. I tend to get numbers mixed up and write things wrong when I do the distributive property. However, it is your preference in the end!(11 votes)

- So, you can apply the distributive property to one of these equations, or you can use the graph/colored boxes thing? Which way is more efficient?(6 votes)
- There is FOIL, double distribution, and box method as 3 ways of doing the exact same thing, they all give the same 4 terms with the middle terms usually combined. FOIL is probably less efficient because it is limited to multiplying two binomials. The box method may require more writing, but might make more sense in exactly what is going on. Double distribution and box method can easily be expanded to larger polynomials in the future. Efficient may be in the eye of the beholder, so find which one works best for you. Many teachers will require you to learn all so that you can make an educated decision which is best, and if you do it enough and have a pretty good math brain, you will be able to do it in your head.(6 votes)

- What would be considered the fastest method of multiplying long polynomials?(2 votes)
- Great question! Yes there is a shortcut, which efficiently combines all the terms for each power of x.

If only one variable, say x, appears in the two polynomials to be multiplied together, then you can use a technique similar to the Vedic multiplication arithmetic technique called vertical and crosswise (which you can look up online to get the idea). Write the terms of each polynomial in order of descending powers of x. For any missing exponents on x, use a coefficient of zero.

Example: multiply (6x^4 + 3x^3 - 5x + 7) by (2x^3 - 4x^2 - x + 8).

Write

(6x^4 + 3x^3 + 0x^2 - 5x + 7)

(0x^4 + 2x^3 - 4x^2 - x + 8).

These two polynomials are now each written with five coefficients.

The idea is to multiply the first coefficient by the first coefficient, then cross multiply the first two coefficients by the first two coefficients, then the first three by the first three, then the first four by the first four, then the first five by the first five, then the last four by the last four, then the last three by the last three, then the last two by the last two, then finally the last one by the last one.

There are no terms of degree 9 or higher.

Coefficient of x^8 is 6(0) = 0.

Coefficient of x^7 is 6(2) + 3(0) = 12.

Coefficient of x^6 is 6(-4) + 3(2) + 0(0) = -18.

Coefficient of x^5 is 6(-1) + 3(-4) + 0(2) + (-5)(0) = -18.

Coefficient of x^4 is 6(8) + 3(-1) + 0(-4) + (-5)(2) + 7(0) = 35.

Coefficient of x^3 is 3(8) + 0(-1) + (-5)(-4) + 7(2) = 58.

Coefficient of x^2 is 0(8) + (-5)(-1) + 7(-4) = -23.

Coefficient of x is -5(8) + 7(-1) = -47.

Constant term (coefficient on x^0) is 7(8) = 56.

The answer is

12x^7 - 18x^6 - 18x^5 + 35x^4 + 58x^3 - 23x^2 - 47x + 56.

Have a blessed, wonderful day!(11 votes)

- is this apart of the algebra 2 course(4 votes)
- I would say this is something you learn in Algebra 1. This seems more like a review in Algebra 2.(5 votes)

- when you are multiplying without a coefficient how do you deal with the exponent.(2 votes)
- Please give an example to clarify. Everything has a coefficient. For example: x = 1x. The coefficient =1.

So, I don't understand what you mean.(9 votes)

- What is the difference between binomials and polynomials(3 votes)
- Binomials have 2 terms and polynomials have 2+ terms so technically binomials are polynomials. They specifically want binomials since they want you to distribute both terms in binomials to the polynomial.(5 votes)

- I got a negative 6 at the last equation and how is it positive 6(0 votes)
- It should be positive 6 because when you distribute the - 6 to the - 1, the product is positive 6.(13 votes)

- what is the different between binomials and polynomials ?(0 votes)
- A binomial is a polynomial with 2 terms.(7 votes)