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## Algebra II (2018 edition)

### Unit 4: Lesson 4

Factoring polynomials - Special product forms- Difference of squares intro
- Factoring using the difference of squares pattern
- Factoring difference of squares: leading coefficient ≠ 1
- Difference of squares
- Factoring perfect squares: negative common factor
- Factoring perfect squares
- Perfect squares
- Factoring using the perfect square pattern
- Factoring difference of squares: two variables (example 2)
- Factor polynomials using structure

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# Factoring difference of squares: leading coefficient ≠ 1

CCSS.Math: , ,

Sal factors 45x^2-125 as 5(3x+5)(3x-5). Created by Sal Khan.

## Want to join the conversation?

- Can 25 can be squared? Yes or no?(25 votes)
- Brian Lee

25 can be squared, and so can any number. Squared means to multiply by itself. When you are finding a square root, not every number can have a square root. There is a helpful song that i know, ahem:

1x1=1

2x2=4

3x3=9

4x4=16

5x5=25********

6x6=36

7x7=49

8x8=64

9x9=81

10x10 is a HUNDRED! EVERY BODY DANCE!

So yes, 25 can be squared and 25 squared will be 625. as i said, any number can be squared. Not every number has a square root. some examples of numbers not have ing a square root is:2,3,5,6,7,8,10, etc. Some number have a square root that is a decimal, but not all. Most do, that arent normally square rotted. Hope this helps :)(30 votes)

- How would you solve a difference of squares for 100g^2 -1(7 votes)
- Sometic,

100g² is (10g)² and

1² is (1)² so both terms are squares. So factor using the difference of squares and you have

(10g+1)(10g-1)

I hope that helps make it click for you.(30 votes)

- what is the purpose of the 5 outside the perenthesis(6 votes)
- The 5 is the Greatest Common Factor that was factored from the polynomial using the distributive property. The new binomial is a difference of 2 squares, so it can be factored to another level using that technique. Thus you create 3 factors: 5 (3x - 5) (3x + 5)(13 votes)

- How do you solve these types of problems when the
**constant**is a**non-square**number?

For example (and don't just solve it!):`96−6x^2`

(5 votes)- First, you must divide by a common factor until both terms are perfect squares, if possible. In this problem, 96 and -6 are both divisible by 6.

After the division, we have 16-x^2. Now you can factor it in the "difference of two squares" method. The square root of 16 is 4, and the square root of x^2 is x.

Important: The factor with the subtraction/negative sign always comes second in the factored form.

Important: Don't forget the factor you divided by! It's the leading coefficient in your final answer.

Knowing this, the factored form of 96-6x^2 is`6(4+x)(4-x).`

I hope this cleared things up.(7 votes)

- why dint he considered sqrt(45)x to power of 2 - sqrt(125) to power of 2 ?(5 votes)
- What does ^ mean?(1 vote)
- And on your calculator you see ^ as well, it does the same thing.

if you have INTL keyboard you can do it the right way: 3² 7³ 5²(3 votes)

- How can factor 3x^2 + 10x + 8 in the forn of (3x ) (x )

or 12x^2 - 17x + 6 = (4x ) (3x )?

Is there a rule to do this? Completing the square will not do.

Where on the Academy can I find the videos for this?

I believe that it is most trial and error, but is a way for trial and least error?

Very, very,very much thanks.

Your advice may also be sent to rainer@dds.nl

Thank you, happy study on the Khan Ac.(3 votes)- You can apply the consequence of Fundamental Theorem of Algebra.

Every quadratic expression of type ax^2 + bx + c can be factored by the form

a(x - x1)(x- x2), where x1 and x2 are the roots of equation.

3x^2 + 10x + 8 = 0

x1 = -4/3

x2 = -2

Then, you get a(x-x1)(x-x2) = 3(x + 4/3)(x+2) = (3x+4)(x+2)

Sorry for my English.(6 votes)

- If this were too be an actual question on a test, which answer would be accepted? Or would both be acceptable? 5(9x^2 - 25) or 5(3x + 5)(3x - 5)?(4 votes)
- That would depend on what the question asked for. If it asks for "fully factored" then 5(3x + 5)(3x - 5) would be correct.

So, you need to know the names of the various forms for expressions so that you know which form a test question is asking for.(3 votes)

- Would (15x+25)(15x-25) be a correct answer due to the distributive property? Would it count as simplying and be a valid answer for a question asking for simplification or would it just be... Unsimplifying again? lol(2 votes)
- Actually that's wrong because you multiplied both factors in parentheses by 5. If you want to distribute the 5, you can multiply it by either of the two factors. Multiplying it by both is like multiplying by 5*5 or 25 - so you just changed the equation.(7 votes)

- How do you factor the difference of two squares?(3 votes)
- Without variables:

25 - 9 = 5² - 3² = 5² + 15 - 15 - 3² = (5 + 3)(5 - 3)

With one variable:

x² - 9 = x² - 3² = x² + 3x - 3x - 3² = (x + 3)(x - 3)

With two variables:

x² - y² = x² + xy - xy - y² = (x + y)(x - y)

When you distribute the first group onto the second, the combined terms cancel out because the two groups have opposite signs and you're left with the first and second terms of each group squared.(2 votes)

## Video transcript

Let's see if we can
factor the expression 45x squared minus 125. So whenever I see
something like this-- I have a second-degree
term here, I have a subtraction
sign-- my temptation is to look at this as a
difference of squares. We've already seen
this multiple times. We've already seen that if
we have something of the form a squared minus b squared, that
this can be factored as a plus b times a minus b. So let's look over here. Well, over here,
it's not obvious that this right over
here is a perfect square. Neither is it obvious
that this right over here is a perfect square. So it's not clear
to me that this is a difference of squares. But what is interesting
is that both 45 and 125 have some factors in common. And the one that
jumps out at me is 5. So let's see if we can factor
out a 5, and by doing that, whether we can get something
that's a little bit closer to this pattern right over here. So if we factor out a 5,
this becomes 5 times-- well, 45x squared divided by 5
is going to be 9x squared. And then 125 divided by 5 is 25. Now, this is interesting. 9x squared-- that's
a perfect square. If we call this a
squared, then that tells us that a
would be equal to 3x. 3x-- the whole thing
squared is 9x squared. Similarly-- I can never say
similarly correctly-- 25 is clearly just 5 squared. So in this case, if we're
looking at this template, b would be equal to 5. So now this is a
difference of squares, and we can factor it completely. So we can't forget our 5 out
front that we factored out. So it's going to be
5 times a plus b. So let me write this. So it's going to be 5 times
a plus b times a minus b. So let me write the b's
down, plus b and minus b. And we're done. 5 times 3x plus 5
times 3x minus 5 is 45x squared minus
125 factored out.