Consider x^2+bx+c = (x+m)(x+n) and expand the product on the right-hand side. Then equate the coefficients. Relate the signs of b and c to the signs of m and n.
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Practice makes perfect
Let's consider a trinomial of the form x^2+bx+c. We are interested in factoring it as the product of two linear terms.
x^2+bx+c = (x+ m)(x+ n)
After expanding the right-hand side product and combining like terms, we get the following equation.
x^2+bx+c &= (x+ m)(x+ n)
&= x^2 + ( m+ n)x + m nNext, we equate the terms of the two quadratic expressions above.
m+ n = b
m n = c
This implies that to factor the original trinomial we have to find factors m and n such that m+ n = b and m n = c. Notice that m and n are factors of c. Next, let's consider the following cases.
If b and c are positive, then m and n are positive.
If b is negative and c is positive, then m and n are negative.
If c is negative, then m and n have opposite signs and the factor with the greater absolute value has the same sign as b.
