Integrated Math 2–3 Honors · Lesson Review · September 24, 2026

Factoring Quadratic Trinomials

Reverse polynomial multiplication by finding factor pairs that create the required middle term—then check every result by multiplying.

27-minute edited lesson11 chaptersCaptions + transcript

01 · Watch

Lesson video

VTT captions SRT captions

0:0001Factoring Quadratic Trinomials 0:0502Reverse Multiplication with Factor Pairs 3:3303A Monic Trinomial: Product and Sum 7:2104Sign Patterns for a Positive Constant 9:2105Opposite Signs for a Negative Constant 11:4606When a Trinomial Does Not Factor 13:2107Special Binomials and a Missing Term 16:0408When the Leading Coefficient Is Not One 18:4109Split the Middle Term and Group 21:2410A Second Leading-Coefficient Example 24:0011Factor Completely: GCF First

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02 · Understand

Learning targets

Factor trinomials of the form $x^2+bx+c$ by finding two numbers with the correct product and sum.
Use signs strategically when the constant or middle coefficient is negative.
Factor trinomials with a leading coefficient by splitting the middle term and grouping.
Check for a GCF first and multiply factors to verify the result.

03 · Connect

Key ideas

Every expression uses KaTeX with semantic MathML for clear visual and assistive-technology reading.

01Reverse multiplication: $(x+3)(x+4)=x^2+7x+12$ Factoring takes a trinomial apart into the binomials that produced it.
02Product and sum: $x^2+6x-27=(x-3)(x+9)$ For $x^2+bx+c$ , find numbers whose product is $c$ and whose sum is $b$ .
03Sign patterns: $x^2-10x+16=(x-8)(x-2)$ A positive constant uses like signs; the middle coefficient tells whether both signs are positive or negative. A negative constant uses opposite signs.
04Leading coefficient: $4x^2+11x+6=4x^2+8x+3x+6$ When $a\ne1$ , multiply $a\cdot c$ , find the required pair, split the middle term, and group.
05Factor completely: $50x^3-140x^2+80x=10x(5x-4)(x-2)$ Always remove a GCF before using another factoring method.

04 · Apply

Worked examples

These examples match the strategies developed in the lesson.

Monic trinomial

Find the pair that multiplies to the constant

x^2+6x-27=(x-3)(x+9)

The numbers $-3$ and $9$ multiply to $-27$ and add to $6$ .

Both signs negative

Use the middle coefficient to select the signs

x^2-10x+16=(x-8)(x-2)

The factors $-8$ and $-2$ multiply to $16$ and add to $-10$ .

Special structure

Watch for a missing term

x^2-25=(x-5)(x+5)

x^2-36x=x(x-36)

A zero middle or constant term often reveals a difference of squares or a shared variable factor.

Leading coefficient

Split the middle term, then group

4x^2+11x+6=4x^2+8x+3x+6

4x(x+2)+3(x+2)=(4x+3)(x+2)

The pair $8$ and $3$ multiplies to $4\cdot6=24$ and adds to $11$ .

Factor completely

GCF before the trinomial method

50x^3-140x^2+80x=10x(5x-4)(x-2)

After removing $10x$ , factor the remaining quadratic and multiply back to check.

05 · Avoid

Common mistakes

Ignoring the GCF.

Before any trinomial method, remove the greatest factor shared by every term.

Checking only the product.

The pair must satisfy both the required product and the required sum.

Losing the sign of the middle term.

Use the middle coefficient to decide which factor has the larger magnitude and which signs are needed.

Forgetting the leading coefficient.

When $a\ne1$ , use the product $ac$ before splitting the middle term.

Stopping after splitting.

Four terms are an intermediate step; finish by grouping and factoring the common binomial.

Skipping the check.

Multiply the binomials to recover the original trinomial exactly.

06 · Check yourself

Try it first

These are new practice problems—not repetitions of the worked examples. Factor each completely before opening its solution.

1Factor

x^2+11x+28

$x^2+11x+28=(x+7)(x+4)$ The pair $7,4$ has product $28$ and sum $11$ .

2Factor

x^2-13x+36

$x^2-13x+36=(x-9)(x-4)$ The pair $-9,-4$ has product $36$ and sum $-13$ .

3Factor

2x^2+9x+4

$2x^2+9x+4=(2x+1)(x+4)$ Since $ac=8$ , split $9x$ as $8x+x$ and group.

4Factor

6x^2-x-2

$6x^2-x-2=(3x-2)(2x+1)$ Since $ac=-12$ , use $-4$ and $3$ to split the middle term.

5Factor

24x^3-36x^2-60x

completely.

$24x^3-36x^2-60x=12x(2x-5)(x+1)$ First remove $12x$ . Then factor $2x^2-3x-5$ .

07 · Revisit

Transcript and captions

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