Integrated Math 2–3 Honors — Factoring Quadratic Trinomials
Corrected lesson transcript


## 0:00 — Factoring Quadratic Trinomials


## 0:05 — Reverse Multiplication with Factor Pairs

[0:05] The factors of the constant term must add to the coefficient of the linear term.
[0:26] We need two factors, called a factor pair.
[0:31] Look for a factor pair of the constant term.
[0:37] What must that factor pair do?
[0:41] The pair must add to the coefficient of the linear term.
[0:52] For this example, use three and four because three plus four is seven.
[1:04] Seven is the coefficient of the linear term.
[1:20] For this example, the factors are two and six.
[1:32] The two numbers multiply to twelve and add to eight.
[1:42] For a negative middle term, negative twelve and negative one multiply to twelve and add to negative thirteen.
[2:00] The first terms of the binomials must multiply to x squared.
[2:10] For this example, use negative three and negative four because they add to negative seven.
[3:05] Both groups now contain the same binomial factor.

## 3:33 — A Monic Trinomial: Product and Sum

[3:33] This is a quadratic trinomial.
[3:40] It's going to be an x here and an x here.
[3:52] Now determine the constant terms in the two binomial factors.
[4:06] List the factor pairs of negative twenty-seven.
[4:44] The sums are twenty-six, negative twenty-six, six, and negative six.
[5:06] The required pair is negative three and positive nine.
[5:10] We need factor pairs of the constant term that add to the coefficient of the linear term.
[5:28] Rewrite six x as negative three x plus nine x, then factor by grouping.
[5:53] The result is x plus nine times x minus three.
[6:27] The factors are the same product in a different order.
[6:32] Multiplication is commutative, so the order does not matter.

## 7:21 — Sign Patterns for a Positive Constant

[7:21] The first term in each binomial is x because x times x is x squared.
[7:48] Before listing pairs, use the sign of the linear coefficient.
[7:53] The coefficient of the linear term is negative. What does that tell us?
[8:00] The constant is positive and the sum is negative, so both factors are negative.
[8:29] The pair negative two and negative eight has sum negative ten.
[8:36] Therefore the factorization is x minus two times x minus eight.
[9:02] Check the factorization by multiplying.
[9:07] The product is x squared minus ten x plus sixteen.

## 9:21 — Opposite Signs for a Negative Constant

[9:21] Try another example.
[9:46] Find factor pairs of negative sixty that add to negative four.
[11:30] A negative constant requires one positive and one negative factor.
[11:41] Their values must differ by four.

## 11:46 — When a Trinomial Does Not Factor

[11:50] Not every polynomial factors over the integers.
[12:42] We are going to learn another method later.
[13:07] The factor pair is positive seven and negative six.

## 13:21 — Special Binomials and a Missing Term

[13:28] We want to recognize these factor pairs quickly.
[13:43] This is a quadratic binomial rather than a quadratic trinomial.
[14:04] Treat the missing linear coefficient as zero.
[14:20] The same method works for a quadratic binomial with no linear term.
[15:01] For this quadratic binomial, factor out the greatest common factor first.
[15:16] Factoring x gives x times x minus thirty-six.
[15:57] Always factor out the greatest common factor before using another method.

## 16:04 — When the Leading Coefficient Is Not One

[16:04] How is this example different?
[16:10] The leading coefficient of the quadratic trinomial is not one.
[16:32] Possible first terms include four x and x, or two x and two x.
[17:08] The constant terms must multiply to positive six.
[17:44] Guess-and-check creates too many possibilities, so use a systematic method.

## 18:41 — Split the Middle Term and Group

[18:41] Four times six is twenty-four.
[18:57] Find factor pairs of twenty-four that add to eleven.
[19:21] Eight and three multiply to twenty-four and add to eleven.
[19:25] Rewrite the middle term as eight x plus three x.
[19:36] This changes eleven x to eight x plus three x without changing the expression.
[19:47] Now there are four terms, so factor by grouping.
[20:13] The result is four x plus three times x plus two.
[20:30] Multiply the leading coefficient by the constant, then find a pair with the required sum.
[20:47] Use the pair to split the middle term, then factor by grouping.

## 21:24 — A Second Leading-Coefficient Example

[21:24] For this example, find factor pairs of four times negative fifteen.
[21:32] The product is negative sixty and the required sum is negative seven.
[22:10] The pair five and negative twelve has sum negative seven.
[22:37] Rewrite negative seven x as five x minus twelve x, then factor by grouping.
[23:25] This method takes more work, but it is systematic.
[23:41] The result is correct, and the order of the two binomial factors does not matter.

## 24:00 — Factor Completely: GCF First

[24:00] This is a cubic trinomial rather than a quadratic trinomial.
[24:08] Before using another method, factor out the greatest common factor.
[24:13] The greatest common factor is ten x.
[24:29] After factoring out ten x, factor the remaining quadratic trinomial.
[26:05] Multiply the factors to check the result.