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Algebra (all content)
Course: Algebra (all content) > Unit 10
Lesson 7: Special products of binomials- Special products of the form (x+a)(x-a)
- Squaring binomials of the form (x+a)²
- Multiply difference of squares
- Special products of the form (ax+b)(ax-b)
- Squaring binomials of the form (ax+b)²
- Special products of binomials: two variables
- More examples of special products
- Polynomial special products: perfect square
- Squaring a binomial (old)
- Binomial special products review
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Squaring a binomial (old)
An old video where Sal expands and simplifies (7x+10)². Created by Sal Khan and Monterey Institute for Technology and Education.
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Where is he getting 140x from?
Thanks.(20 votes)
49x^2+70x+70x+100, simply add the 70x+70x = which is 140x(5 votes)
I thought that with powers to powers you distributed the exponent to the exponents inside the parenthesis? Hence...
(x+3)^2=x^2+3^2=x^2+9
Why is this not true?(5 votes)
Because:
1.) We need to follow the order of operations P.E.M.D.A.S
2.) A set of numbers inside a parenthesis is considered as a single number itself.
To see how it works using numbers instead of variables may make things clearer for us.
Lets substitute x by 5 and see what happens when we follow P.E.M.D.A.S.
(5+3)^2
=25+2(5)3+9
or 8^2 → I followed the order of operations P.E.M.D.A.S, I first did the operation inside the parenthesis then I distributed the exponent.
= 64
Doing the wrong way would lead to a different answer.
(5+3)^2
=5^2+3^2
=25+9
=34(6 votes)
what does he mean when he says( 7x*10)*2?(4 votes)
Are you using * as a multiplication sign?(3 votes)
- find the product (s-1)(s+1)(3 votes)
Two options here: Either you know your rules for binomials and write down (s^2-1) or you use the distributive property twice:(s-1)(s+1)
= s(s+1) - 1(s+1)
= s^2 + s - s - 1
= s^2 - 1(3 votes)
- how do I do (x^2)^2? Is it x^5? Do we add the exponents or multiply them?(3 votes)
When you have an expression in the form (a^b)^c, you can simplify it to a^bc.
In your example, (x^2)^2= x^(2*2)= x^4.
This is one of the exponent properties, and you can learn more about it in here-
https://www.khanacademy.org/math/algebra/exponent-equations/exponent-properties-algebra/v/exponent-properties-3(3 votes)
Sooo, 900x+900x would equal 1,800x. Am I right?(3 votes)- need to know the differnce of squares(2 votes)
- Any time a binomial has 2 terms which are perfect squares subtracted from each other, it is a difference of squares, and it factors like this:
a^2 - b^2 = (a+b)(a-b)
Examples:
x^2 - 4 = (x+2)(x-2)
a^2 - 16 = (a+4)(a-4)
4y^2 - 9q^2 = (2y + 3q)(2y - 3q)
25x^2 - 1 = (5x+1)(5x-1)(4 votes)
how did he distribute it to that long expression?(1 vote)- Okay, so!
(7x+10)^2
You have (7x+10) BUT You have the exponent of 2, so that means you're multiplying, when multiplying polynomials you must simplify like regular multiplication
Which is multiplying it by itself because it's squared
Meaning you are left with (7x+10) times (7x+10)
So you take the 7x out of the (7x+10) and multiply it by the remaining (7x+10), because when something is squared you are multiplying by itself.
Therefore 7x(7x+10) would make 49X^2 + 70x
because 7*7 is 49 but you still have the x, and X and another X is the same as X^2. Now again you multiply 7x by 10 and get 70x, because the X just doesn't disappear.
Now, you multiplied one factor with the equation, but what about the +10? So now we just multiply 10 times the same equation!
So 10(7x+10), 10*7x is 70x because you are still multiplying the X by the coefficient, so a constant times a coefficient does nothing to the variable. Now you multiply 10 times 10 to get 100, leaving you with 70x+100
Now, you are left with 49x^2+70x+70x+100, just combine like terms, which would be 70x and 70x because they are both real numbers and share the variable X, Leaving you with 49x^2 + 140x + 100
I hope this helps you, it gets easier when you practice. send me a message if you need anymore help.(3 votes)
- (a+b)^3 , I wanna know how to do this step by step. I dont know why they answer is the way it is. Not just the answer but how to do step by step?(1 vote)
- (a+b)^3
= (a+b)(a+b)(a+b)
Use FOIL rule to multiply the last 2 binomials. Do you know FOIL?
= (a+b)(a^2 + 2ab + b^2)
= a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2)
= (a^3 + 2a^2b + ab^2) + (a^2b + 2ab^2 + b^3)
= a^3 + 2a^2b + a^2b + ab^2 + 2ab^2 + b^3
= a^3 + 3a^2b + 3ab^2 + b^3(3 votes)
So I was solving problems in the "Multiplying Binomials by Binomials Practice" and I came across this one problem that I didn't understand.
The problem was as follows:(x - 7)(x - 5) = x^2 - 12x + d
I simplified the equation to:x^2 - 5x - 7x + 35
But I don't understand how 5x - 7x = 12x. Shouldn't 5x - 7x be something like -2x?(1 vote)
Oh you didn't look at the equation correctly! It's actually supposed to be -5x - 7x, always take the negative sign in front of it. And it's supposed to be -12x not 12x. So -5x - 7x = -12x.
Hope this helped :D
Pssstt...by the way, I love the username!!(2 votes)
Video transcript
We're asked to simplify
7x plus 10 squared. And your temptation when you see
something like this might be to just make it 7x squared
plus 10 squared and that would be incorrect. Remember, when you're squaring
something, you're multiplying something by itself. So this wouldn't be 7x squared
plus 10 squared. This is equal to 7x plus
10 times 7x plus 10. This is not equal to-- sometimes
if you said this is 7x times 10 squared, then you
could say this is equal to 7x squared times 10 squared. You would be able to do it in
this situation right here. But that's not what's
going on here. This is 7x plus 10 squared,
which is just 7x plus 10 times 7x plus 10. So let me delete
that up there. That's probably the single
biggest point of confusion when people first learn
to take a square of an expression like this. So always remember that it's 7x
plus 10 times 7x plus 10. And now we can multiply
it out. So one, we could distribute this
yellow 7x plus 10 onto each of the terms into
the green one. So it would be 7x plus 10
times 7x right here. So we put the 7x out front. And then we could say plus
7x plus 10 times 10. Or we could say plus 10
times 7 x plus 10. I just distributed the 7x plus
10 on each of these terms. Then you do the distributive
property again. 7x times 7x is 49x squared. 7x times 10 is 70x. And then you have 10 times
7x is another 70x. And then you have 10 times
10, which is 100. Now, there's something
interesting you might want to notice right now is that this
expression, it is the first expression squared, right? 7x, the whole expression
squared is 49x squared. And then you do have your
last expression squared, 10 squared is 100. But then in between, you have
the product of these two things twice, and that's just
a byproduct of having to do all of the different
combinations of products. So the end expression, when you
simplify everything, it becomes 49x x squared
plus 140x plus 100. So if you want to remember
a quick way of squaring a binomial like this, and just
remember, it is coming out of this distributive property, but
if you had to do it really fast, it's just going to be this
first term squared, which is that right there, plus 2
times the product of these two terms. 7x times 10 is 70x,
and then you multiply it by 2, you get 140x. You get that right there. And then the last term is just
going to be the square of the second term right there. 10 squared is 100.