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### Course: Class 9 > Unit 9

Lesson 5: Multiplying polynomials# Multiplying binomials by polynomials

Learn how to multiply binomials by polynomials with ease! This lesson breaks down the process into simple steps, using the distributive property to multiply each term. You'll master combining like terms to simplify expressions, turning complex polynomials into manageable math. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- How did Sal get 50a^3 in this video? Around1:07into the video he gets 50a^3 for his answer. Did he add the variable a and three together?(6 votes)
- To multiply terms with variables in them (for instance 10a times 5a^2), you have to :

1. Multiply the numerical coefficients : in this case, 10 times 5 = 50

2. Look for the same variable : in this case, a times a^2.

3. Write the variable with an exponent that is the sum of the exponents :

in this case, 1 + 2, giving a^3.

4. So the answer to 10a times 5a^2 is 50 a^3.

Hope this has been helpful.(22 votes)

- Whats the hardest polynomial that anyone has encountered?(6 votes)
- SIMPLIFY

(2345.7647x^56 + 66t^5353)(6433x^343443)(35535t^3553+ 434x^3)

Edit: I swear, if that David guy solves this-(6 votes)

- Does it matter which way you FOIL out(8 votes)
- No, you will get the same answer despite which way you start.(2 votes)

- Hey whenever Sal was doing the problem in the video and he subtracted -10a-21a how come he didn't make it -31a^2? Just wondering. :)(4 votes)
- he wasn't multiplying the polynomials he added them so the exponents stay the same because when you have -10 of something and -21 of something the something doesn't change.(6 votes)

- There is no way faster that I know of... But if you find one, Let me know! LOL(5 votes)
- Of course. Faster-enter the numbers in your calculator. Maybe just as good, round numbers and use scientific notation to approximate. Learning should not be a race. Life is over soon enough.(6 votes)

- Wouldn't Sal need to multiply -10a by -15a^2 before trying to merge terms?(4 votes)
- Maybe writing out all the steps will help.

(10a-3)(5a² + 7a - 1) First he distributed the bigger set of parenthesis into the smaller set.

10a(5a² + 7a - 1) - 3(5a² + 7a - 1) Notice how the two terms are still being subtracted. In other words if (5a² + 7a - 1) was replaced with x it would look like 10ax - 3x. Now there is no multiplication between the two different parts with an x. Now do the rest of the distributions.

50a³ + 70a² - 10a - 15a² - 21a + 3 Sal showed what to do from here. the -3 was distrbuted into the right side, but the right side was being subtracted from the left, so there is no multiplication.

let me know if this did not help, I can try a different way of explaining.(7 votes)

- Is there a faster way of doing it lol?(6 votes)
- Not really

But for (a + b)(c + d) where b and d are numbers

It will be a^2 + (b+d)x + (b*d)

SO, (x+3)(x+7) x^2 + 10x + 21(1 vote)

- I dont understand polynomials(2 votes)
- you will eventually ☺️(7 votes)

- Are there other ways to solve this?(4 votes)
- No, not really. I mean you could do something like

(10a - 3)(5a² + 7a - 1) = (10a - 3)5a² + (10a - 3)7a - (10a - 3) = ...

or

(10a - 3)(5a² + 7a - 1) = 50a³ + 70a² - 10a - 15a² - 21a + 3 = ...,

but at the end of the day you are basically doing the same thing.(2 votes)

- I understand how to multiply polynomials but what about when they ask to simply an equation like this to get it's standard form:

-2 (p+4)^2 -3+5p

What are the steps to solve an equation like this?(2 votes)- 1) You do not have an equation. An equation is made up on 2 expressions separated by and equals symbol. What you have is a polynomial expression.

2) You can simplify your expression. Follow PEMDAS rules.

-- Deal with the exponent: FOIL (p+4)^2

-- Distribute the -2 across the results of the FOIL. This will eliminate the parentheses.

-- Combine like terms.

-- Order the terms from highest degree to lowest degree to get your answer into standard form.

Hope this help.(4 votes)

## Video transcript

We are multiplying 10a minus
3 by the entire polynomial 5a squared plus 7a minus 1. So to do this, we can just
do the distributive property. We can distribute this
entire polynomial, this entire trinomial,
times each of these terms. We could have 5a squared
plus 7a minus 1 times 10a. And then 5a squared plus 7a
minus 1 times negative 3. So let's just do that. So if we have-- so let
me just write it out. Let me write it this way. 10a times 5a squared
plus 7a minus 1. That's that right over here. And then we can have
minus 3 times 5a squared plus 7a minus 1. And that is this
distribution right over here. And then we can simplify it. 10a times 5a squared--
10 times 5 is 50. a times a squared
is a to the third. 10 times 7 is 70. a times a is a squared. 10a times negative
1 is negative 10a. Then we distribute this
negative 3 times all of this. Negative 3 times 5a squared
is negative 15a squared. Negative 3 times
7a is negative 21a. Negative 3 times
negative 1 is positive 3. And now we can try
to merge like terms. This is the only a to
the third term here. So this is 50a to the third. I'll just rewrite it. Now we have two a squared terms. We have 70a squared minus
15, or negative 15a squared. So we can add these two terms. 70 of something minus
15 of that something is going to be 55
of that something. So plus 55a squared. And then we also
have two a terms. We have this negative 10a, and
then we have this negative 21a. So if we go negative 10 minus
21, that is negative 31. That is negative 31a. And then finally, we only have
one constant term over here. We have this positive 3. So plus 3. And we are done.