➕ Algebra 1 · Factoring
Factoring Trinomials: true or false
Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.
All statements and answers
x2 + 9x + 18 = (x + 3)(x + 6).
✓ True. 3 · 6 = 18 and 3 + 6 = 9. Check: (x + 3)(x + 6) = x2 + 6x + 3x + 18 = x2 + 9x + 18.
x2 − 6x + 8 = (x + 2)(x + 4).
✗ False. (x + 2)(x + 4) expands to x2 + 6x + 8, with +6x. Because b = −6 and c = 8 is positive, both numbers must be negative: x2 − 6x + 8 = (x − 2)(x − 4).
x2 + 4x − 12 = (x + 6)(x − 2).
✓ True. 6 · (−2) = −12 and 6 + (−2) = 4. Check: (x + 6)(x − 2) = x2 − 2x + 6x − 12 = x2 + 4x − 12.
x2 + 5x − 14 = (x − 7)(x + 2).
✗ False. (x − 7)(x + 2) expands to x2 − 5x − 14, with −5x. The pair needs a sum of +5, so the larger number must be positive: x2 + 5x − 14 = (x + 7)(x − 2).
If c is positive in x2 + bx + c, the numbers p and q in (x + p)(x + q) must have the same sign.
✓ True. pq = c, and a product of two integers is positive only when they have the same sign (both positive or both negative).
x2 + 25 = (x + 5)(x + 5).
✗ False. (x + 5)(x + 5) expands to x2 + 10x + 25. For x2 + 25 you would need p + q = 0 and pq = 25, which is impossible with integers (same signs cannot add to 0). So x2 + 25 is prime.
(2x + 6)(x + 3) is a completely factored form of 2x2 + 12x + 18.
✗ False. It expands correctly, but 2x + 6 = 2(x + 3), so a common factor was left in. The complete answer is 2(x + 3)(x + 3), which is 2(x + 3)2.
Every trinomial x2 + bx + c with integer b and c can be factored into two binomials with integer coefficients.
✗ False. Some are prime. For x2 + 2x + 6, the pairs with product 6 are 1 · 6 (sum 7) and 2 · 3 (sum 5), and the negative pairs have negative sums. None adds to 2.
To factor 4x2 + 11x + 6 with the ac method, look for two numbers with product 6 and sum 11.
✗ False. With a ≠ 1 you need product ac = 4 · 6 = 24 and sum 11. That pair is 3 and 8, which gives (4x + 3)(x + 2). Check: 4x2 + 8x + 3x + 6 = 4x2 + 11x + 6.
16x2 + 8x + 1 is a perfect square trinomial.
✓ True. 16x2 = (4x)2, 1 = 12, and 2 · 4x · 1 = 8x is the middle term. So it equals (4x + 1)2.
x2 + 10x + 9 is a perfect square trinomial because 9 is a perfect square.
✗ False. The middle term would have to be 2 · x · 3 = 6x, not 10x. It is not a perfect square trinomial. It factors as (x + 1)(x + 9).
The first step in factoring 5x2 + 20x + 15 is to take out the GCF, 5.
✓ True. 5x2 + 20x + 15 = 5(x2 + 4x + 3) = 5(x + 1)(x + 3). Taking out the 5 first leaves a trinomial with a coefficient of 1 on x2.
(x − 4)(x + 5) = 0 when x = 4 or x = −5.
✓ True. A product is 0 only when a factor is 0. x − 4 = 0 gives x = 4, and x + 5 = 0 gives x = −5.
The completely factored form of 7x2 + 14x + 21 is 7(x2 + 2x + 3).
✓ True. The GCF is 7. Inside, the pairs with product 3 are 1 · 3 (sum 4) and (−1)(−3) (sum −4). Neither sum is 2, so x2 + 2x + 3 is prime and nothing more can be done.