We are given a quadratic trinomial of the form ax^2+bx+c, where |a| ≠1 and there are no common factors. We will try to factor this expression by rewriting the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b.
3x^2-8x+15 ⇔ 3x^2+(- 8)x+15We have that a= 3, b=- 8, and c=15. There are now three steps we need to follow in order to rewrite the above expression.
Find a c. Since we have that a= 3 and c=15, the value of a c is 3* 15=45.
Find factors of a c. Since a c=45, which is positive, we need factors of a c to have the same sign — both positive or both negative — in order for the product to be positive. Since b=- 8, which is negative, those factors will need to be negative so that their sum is negative.
c|c|c|c
1^(st)Factor
&2^(nd)Factor
&Sum
&Result
-1
&- 45
&-1 + (-45)
&- 46
-3
&- 15
&-3 + (-15)
&- 18
-5
&- 9
&-5 + (-9)
&- 14
Because the pairs of factors of 45 do not add up to - 8, the expression cannot be factored using integers. Therefore, it is a prime polynomial.
