Integrated Math 2–3 Honors — Factoring Differences of Squares
Corrected lesson transcript


## 0:00 — Recognize a Missing Linear Term

[0:05] Let us factor this expression. To factor means to write it as a product.
[0:12] What do we call this polynomial?
[0:18] It is a quadratic binomial.
[0:25] Is it missing the quadratic, linear, or constant term?
[0:33] It is missing the linear term.
[0:36] We could write it like this.
[0:44] We need a factor pair whose product is negative 49 and whose sum is zero.
[0:55] The pair is negative seven and positive seven.
[0:56] plus seven.
[1:02] That's plus seven.
[1:04] That gives x minus seven and x plus seven.
[1:09] Yes. Easy peasy. Wow.
[1:13] Okay.
[1:16] Do we need to write this step? Definitely not.

## 1:19 — Irrational Factors Still Work

[1:19] We need two numbers whose product is negative five and whose sum is zero.
[1:22] to be negative 5 and add to be 0.
[1:34] Maybe it's not.
[1:37] Maybe it's not.
[1:41] Definitely not.
[1:44] What did we...
[1:56] Somebody said... What did somebody say?
[1:59] What did you say?
[2:02] Uh...
[2:02] Rad...
[2:03] Use square root of five and negative square root of five.
[2:04] I'm gonna be a bad man.
[2:06] Uh...
[2:07] Rad pod and negative red pod.
[2:13] Those numbers multiply to negative five and add to zero.
[2:21] These factors are not integers, but they are valid real numbers.
[2:25] Integers, but they are two numbers that multiplied to be negative five and add to be zero
[2:32] The factorization is x minus square root of five times x plus square root of five.
[2:39] Should we check and see if it works?
[2:44] Multiply to check: the middle terms cancel, and the product is x squared minus five.
[3:01] Oops, I normally do it here.
[3:02] here. And we get minus 5
[3:10] at 0
[3:11] and that's it.
[3:16] Okay.

## 3:16 — Sum or Difference of Squares?

[3:16] Can two real numbers multiply to 36 and add to zero?
[3:31] It's not 6 and negative 6.
[3:33] Why not?
[3:36] Yeah, that does not multiply to be 36.
[3:42] Are there any?
[3:46] There are none.
[3:48] They would need the same sign to have a positive product, so their sum cannot be zero.
[3:55] Two positives add together, never will be zero. Two negatives add together will never be zero.
[4:01] Therefore this expression does not factor over the real numbers.
[4:08] This is called a sum of squares.
[4:15] A sum of squares does not factor over the real numbers.
[4:18] Okay.

## 4:18 — The Difference-of-Squares Identity

[4:18] The earlier examples are differences of squares.
[4:21] What do we call this?
[4:22] The difference of squares.
[4:23] Difference.
[4:28] Oh, squares.
[4:34] Okay.
[4:34] A difference of two squares always factors.
[4:44] A squared minus b squared factors as a minus b times a plus b.
[4:47] always factors to what?
[4:54] A plus B times
[4:56] A minus B.
[4:59] and that order doesn't matter.
[5:06] Let's call the
[5:08] difference of squares.
[5:33] Write x squared minus one hundred as x squared minus ten squared.
[5:43] It factors as x minus ten times x plus ten.
[5:48] OK.

## 5:48 — Perfect-Square Coefficients

[5:48] Rather than multiplying everything out, recognize the square terms.
[5:52] and then look for two factors and...
[5:56] Too complicated.
[5:58] It's very easy.
[6:00] Four x squared is the square of two x.
[6:03] 2x.
[6:05] Thirty-six is the square of six.
[6:07] Six.
[6:09] This is a difference of squares.
[6:11] It factors as two x minus six times two x plus six.
[6:13] 2x minus six.
[6:15] That's 2x plus six.
[6:17] Done.
[6:17] Let's try another one.
[6:37] This expression is a difference of squares.
[6:45] One hundred x squared is the square of ten x, and eighty-one is the square of nine.
[6:50] 81 is 9.
[6:56] It factors as ten x minus nine times ten x plus nine.
[7:03] plus 9

## 7:13 — Two Variables and Irrational Squares

[7:13] Uh...
[7:25] The square roots are four x and five y.
[7:30] So the factors are four x minus five y and four x plus five y.
[7:37] Done.
[7:37] Okay, let's try another one.
[8:00] This is also a difference of squares.
[8:01] The square root of this?
[8:08] The square root of twenty-five x squared is five x.
[8:12] What's the square root of this?
[8:14] The square root of seven stays square root of seven, giving conjugate factors.

## 8:25 — Factor Higher Powers Completely

[8:25] This is a fourth-degree binomial rather than a quadratic binomial.
[8:27] What would this be?
[8:29] 4th degree bound.
[8:38] However, it is still a difference of squares.
[8:48] The square root of x to the fourth is x squared, and the square root of sixteen is four.
[8:50] x squared.
[8:53] What squared is 16?
[8:55] 4.
[8:55] 4. This is going to be
[8:59] First factor as x squared minus four times x squared plus four.
[9:03] squared plus 4.
[9:08] We are not finished because x squared minus four is another difference of squares.
[9:13] Factor x squared minus four again.
[9:14] difference of squares.
[9:19] The complete factorization is x minus two times x plus two times x squared plus four.
[9:32] The sum of squares x squared plus four is prime over the real numbers.
[9:41] Difference of squares lets us factor higher-degree polynomials.
[9:54] And it's going to be very useful when we learn something else later.

## 9:57 — Repeat the Pattern

[9:57] But let's try one more.
[10:06] This is another difference of squares.
[10:15] What square is x need?
[10:19] The square root of x to the eighth is x to the fourth, and the square root of one is one.
[10:32] And we have two fourth degree polynomials. Is one of the two difference the squares?
[10:38] Factor x to the fourth minus one again as x squared minus one times x squared plus one.
[11:07] We are not done because x squared minus one is another difference of squares.
[11:10] No. We got another
[11:14] difference of squares.
[11:29] Now the expression is completely factored.
[11:35] What's that?
[11:37] The result is the product of four binomials, and their order does not matter.
[11:39] 4 binomials. The order does not matter.
[11:43] Because the multiplication is commuted. Sorry, let's do one more.

## 11:47 — Greatest Common Factor First

[11:47] First factor out the greatest common factor.
[11:51] Factor out what?
[11:53] Always remove the greatest common factor before applying another factoring method.
[11:58] What is the greatest common factor?
[12:05] The greatest common factor is ten x cubed.
[12:15] The remaining quadratic binomial is a difference of squares.
[12:42] Okay, we good?