Find two numbers that multiply to one thing and add to another. That is the whole method.
Factor
x² − 5x − 24
We need two numbers that multiply to −24 and add to −5: those are −8 and +3, giving (x − 8)(x + 3). B has the signs the wrong way round — 8 and −3 multiply to −24 but add to +5. C and D multiply to −24 but add to −2 and −10. Check by expanding: x² + 3x − 8x − 24 = x² − 5x − 24 ✓
For x² + bx + c, find m and n with mn = c and m + n = b.
x² + 11x + 24
need product 24, sum 11
1×24, 2×12, 3×8 ✓, 4×6
list the factor pairs
3 + 8 = 11
the pair we want
= (x + 3)(x + 8)
done
Listing the factor pairs systematically is faster than guessing, and it proves you have not missed one.
| c | b | The two numbers are | Example |
|---|---|---|---|
| + | + | both positive | x² + 7x + 12 → 3, 4 |
| + | − | both negative | x² − 7x + 12 → −3, −4 |
| − | either | one of each; the bigger one takes b's sign | x² − 5x − 24 → −8, 3 |
Reading the signs first cuts the search in half. If c is positive, you never need to consider a mixed pair at all.
Multiply a by c, find the pair for that, then split the middle term and group.
3x² + 11x + 6
a = 3, c = 6
ac = 18, need sum 11
2 and 9
3x² + 2x + 9x + 6
split the middle
x(3x + 2) + 3(3x + 2)
group in pairs
= (3x + 2)(x + 3)
common bracket out
Looking for a pair that multiplies to 6 instead of 18 is the standard error here. When a ≠ 1, the target product is a × c.
a² + 2ab + b² = (a + b)² · a² − 2ab + b² = (a − b)²
4x² − 12x + 9
first and last are squares
√4x² = 2x, √9 = 3
the candidates
2(2x)(3) = 12x ✓
the middle term matches
= (2x − 3)²
negative middle → minus
4x² − 10x + 9 has square ends too, but 2(2x)(3) is 12x, not 10x — so it is not a perfect square. Both conditions must hold.
2x² + 10x + 12
every term is even
= 2(x² + 5x + 6)
now a = 1, much easier
= 2(x + 2)(x + 3)
2 and 3
A negative ac means one positive and one negative number; the bigger one takes b's sign.
Factor 4x² − 4x − 15.
ac = 4 × (−15) = −60, need sum −4
one of each sign
−10 and 6
−10 × 6 = −60, −10 + 6 = −4
4x² − 10x + 6x − 15
split the middle
2x(2x − 5) + 3(2x − 5)
group in pairs
= (2x − 5)(2x + 3)
common bracket out
10 and −6 also multiply to −60, but they add to +4. Using them gives (2x + 5)(2x − 3), which expands to 4x² + 4x − 15 — the middle sign is flipped. Always check the sum, not just the product.
Factor x⁴ − 10x² + 9 completely.
let u = x²: u² − 10u + 9
a trinomial in u
= (u − 1)(u − 9)
−1 × −9 = 9, −1 + −9 = −10
= (x² − 1)(x² − 9)
put x² back
= (x + 1)(x − 1)(x + 3)(x − 3)
each bracket is a difference of squares
When the powers go x⁴, x², number, treat x² as one unknown. Then re-check every bracket, as in "factor completely".
The same pair hunt, in three new settings.
x² − 2xy − 35y²
need product −35, sum −2
−7 and 5
= (x − 7y)(x + 5y)
each number carries a y
Check the middle: 5xy − 7xy = −2xy ✓. The y goes with the number, since y · y = y².
−x² + 3x + 28
= −(x² − 3x − 28)
take out −1 first
= −(x − 7)(x + 4)
−7 × 4 = −28, −7 + 4 = −3
A negative leading term is easiest removed as a common factor of −1, which flips every sign inside.
x² + 4x + 6
pairs of 6: 1 and 6, 2 and 3
sums 7 and 5
no pair adds to 4
the list is complete
The trinomial does not factor over the integers. Listing every pair is what lets us say so with confidence.
Items ask you to factor, to pick which binomial is a factor, or to spot a perfect square trinomial. The fastest route is GCF first, read the signs, then list the factor pairs of c (or of ac). For a 10-second check, add the outer and inner products of your answer — they must give the middle term.
Factor
2x² − 7x + 3
ac = 2 × 3 = 6, and we need a pair multiplying to 6 and adding to −7: that is −1 and −6. Splitting: 2x² − x − 6x + 3 = x(2x − 1) − 3(2x − 1) = (2x − 1)(x − 3). Check: 2x² − 6x − x + 3 ✓ B has the signs mixed and expands to 2x² − 5x − 3.
Hint: c is positive and b is negative, so both numbers must be negative. That rules out every option with a plus sign in it.
