The factorization of the type x trinomial2 + Sx + P is the 4th case of factorization that comes right after the trinomial of the perfect square, as it is also used when the algebraic expression is a trinomial.
When it is necessary to factor an algebraic expression and this is a trinomial (three monomials), and we verified that this does not form a trinomial of the perfect square, so we must use factorization type x2 + Sx + P.
Given the algebraic expression x2 + 12x + 20, we know it's a trinomial, but its two end members are not squared, so it rules out the possibility of it being perfect square. So the only factorization case we can use to factor this algebraic expression is x2 + Sx + P. But, how are we going to apply this factorization in the expression x2 + 12x + 20? See the resolution below:
We should always look at the coefficients of the last two terms, see:
x2 + 12x + 20. The numbers 12 and 20 are the coefficients of the last two terms, now we must find two numbers that when we add the value will be equal to + 12 and when we multiply the result will be equal to + 20, we will reach these numbers through attempts.
The added and multiplied numbers that give the value 12 and 20, respectively, are 2 and 10.
2 + 10 = 12
2. 10 = 20
So we factored using the numbers found which in the example is 2 and 10, so the factored form ofx2 + 12x + 20 it will be (x + 2) (x + 10).
See some examples that use the same line of reasoning as the example above:
Example 1
x2 – 13x +42, to factor this algebraic expression we must find two numbers whose sum equals -13 and its product equals 42. These numbers will be -6 and -7, because: - 6 + (- 7) = -13 and – 6. (- 7) = 42. Therefore, the factorization will be equal to:
(x – 6) (x – 7).
by Danielle de Miranda
Graduated in Mathematics
Brazil School Team
Algebraic Expression Factorization
Math - Brazil School
Source: Brazil School - https://brasilescola.uol.com.br/matematica/trinomio-tipo-x-sx-p.htm