LEMMA

Polynomials & quadratics

Factor quadratic trinomials

ALEKS placement

Find two numbers that multiply to the constant and add to the middle coefficient.

What this covers

x2+bx+c=(x+p)(x+q)pq=cp+q=b​
  • Take out any factor every term shares first. That often makes the leading coefficient 1.
  • List the factor pairs of the constant from 1 upward. Stop at the pair that adds to the middle coefficient.
  • Read the signs before you search: a positive constant means both numbers take the middle term's sign.
  • A negative constant needs opposite signs. The number with greater absolute value takes the middle term's sign.
  • If the leading coefficient is not 1, multiply a·c, find the pair for that product, split the middle term, and group.
  • After grouping, the two brackets must match; if not, recheck your pair. Multiply back out to check the answer.

Worked example

Worked example

Factor x² − 7x + 12

  1. (−3)(−4)=12The constants in the two groups must multiply to twelve. Two negative numbers give a positive product.
  2. (−3)+(−4)=−7They must also add to negative seven, the coefficient of the middle term. This pair passes both checks.
  3. x2−7x+12=(x−3)(x−4)Use the pair to build two factors. Factoring reverses multiplication.
  4. (x−3)(x−4)=x2−4x−3x+12Check by multiplying each term in the first group by each term in the second.
  5. x2−4x−3x+12=x2−7x+12The two middle products combine to the original middle term. All three terms match.
Another worked example

Factor x² + x − 12 over the integers.

  1. 4(−3)=−12The two constants must multiply to the final number. Opposite signs give a negative product.
  2. 4+(−3)=1Their sum must also match the middle coefficient. This pair passes both checks.
  3. x2+x−12=(x+4)(x−3)Expansion gives x² − 3x + 4x − 12.

A common mistake

A mistake Lemma catches

From x2−7x+12

x2−7x+12=(x+3)(x−4)x2−7x+12=(x−3)(x−4)

“(x + 3)(x − 4) pairs a positive number with a negative one, and their product is (3)(−4) = −12, but the constant is +12. A positive constant with a negative middle term needs two negative numbers: (−3)(−4) = 12 and (−3) + (−4) = −7.”

Try one

Sample problem

Factor completely.

x2+12x+32

Practice factor trinomials free

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