Factoring with Positive Terms

Factor: $x^2 + 7x + 12$

Identify what we need: Find two numbers that ADD to 7 and MULTIPLY to 12 = Sum = 7, Product = 12

List factor pairs of 12: $1 \times 12$, $2 \times 6$, $3 \times 4$ = Three possibilities

Check which pair sums to 7: $1 + 12 = 13$ (no), $2 + 6 = 8$ (no), $3 + 4 = 7$ (yes!) = $m = 3$, $n = 4$

Write the factored form: $(x + 3)(x + 4)$ = $(x + 3)(x + 4)$

Verify by expanding: $x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ \checkmark = Correct!

Answer: $(x + 3)(x + 4)$

Factoring with a Negative Middle Term

Factor: $x^2 - 9x + 20$

Identify sum and product: Sum = $-9$, Product = $20$ = Both factors must be negative (negative sum, positive product)

List negative factor pairs of 20: $(-1)(-20)$, $(-2)(-10)$, $(-4)(-5)$ = Three possibilities

Check which pair sums to -9: $-1 + (-20) = -21$, $-2 + (-10) = -12$, $-4 + (-5) = -9$ (yes!) = $m = -4$, $n = -5$

Write the factored form: $(x - 4)(x - 5)$ = $(x - 4)(x - 5)$

Verify by expanding: $x^2 - 5x - 4x + 20 = x^2 - 9x + 20$ \checkmark = Correct!

Answer: $(x - 4)(x - 5)$

Factoring with a Negative Constant

Factor: $x^2 + 2x - 15$

Identify sum and product: Sum = $2$, Product = $-15$ = One factor positive, one negative (negative product)

List factor pairs of 15: $1 \times 15$, $3 \times 5$ = Consider both signs

Find the combination that sums to 2: $5 + (-3) = 2$ (yes!) or $-5 + 3 = -2$ (no) = $m = 5$, $n = -3$

Write the factored form: $(x + 5)(x - 3)$ = $(x + 5)(x - 3)$

Verify by expanding: $x^2 - 3x + 5x - 15 = x^2 + 2x - 15$ \checkmark = Correct!

Answer: $(x + 5)(x - 3)$

Factoring with Both Negative Terms

Factor: $x^2 - 3x - 28$

Identify sum and product: Sum = $-3$, Product = $-28$ = Different signs (negative product), larger is negative (negative sum)

List factor pairs of 28: $1 \times 28$, $2 \times 14$, $4 \times 7$ = Consider both signs

Find the combination that sums to -3: $4 + (-7) = -3$ (yes!) = $m = 4$, $n = -7$

Write the factored form: $(x + 4)(x - 7)$ = $(x + 4)(x - 7)$

Verify by expanding: $x^2 - 7x + 4x - 28 = x^2 - 3x - 28$ \checkmark = Correct!

Answer: $(x + 4)(x - 7)$

Mistake: Confusing sum and product requirements

Why: Students sometimes look for numbers that multiply to $b$ and add to $c$, when it should be the opposite.

Correct: Remember: the numbers ADD to the middle coefficient ($b$) and MULTIPLY to the constant ($c$).

Mistake: Forgetting to consider negative factors

Why: When $c$ is positive but $b$ is negative, both factors must be negative.

Correct: Use the sign rules: positive product means same signs, negative product means different signs.

Mistake: Writing $(x + 3)(x + 4)$ as $x^2 + 34$

Why: This error comes from adding the constants instead of properly expanding.

Correct: Always verify by FOILing: $(x+3)(x+4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$

Mistake: Assuming all trinomials can be factored with integers

Why: Some trinomials are prime (cannot be factored with integers).

Correct: If no integer pair works, the trinomial is prime. Example: $x^2 + 5x + 3$ has no integer factors.

Projectile Motion

When an object is thrown upward, its height follows a quadratic pattern. Factoring helps find when it hits the ground.

A ball's height is given by $h = -t^2 + 5t + 6$. Factoring $-1(t^2 - 5t - 6) = -(t-6)(t+1)$ shows it lands at $t = 6$ seconds.

Garden Design

Landscape architects use factoring to determine dimensions when given area constraints.

A garden's area is $x^2 + 11x + 24$ square meters. Factoring gives $(x + 3)(x + 8)$, revealing possible dimensions.

Q: What if I cannot find two numbers that work?

A: The trinomial may be prime (cannot be factored with integers). For example, $x^2 + 5x + 3$ has no integer factor pairs that add to 5 and multiply to 3.

Q: Does the order of the binomials matter?

A: No! $(x + 3)(x + 4)$ is the same as $(x + 4)(x + 3)$ due to the commutative property of multiplication.

Q: What if the leading coefficient is not 1?

A: That requires a different method (AC method or trial and error). This lesson focuses only on trinomials where $a = 1$.