Question

factor trinomials (a=1)
score: 2/5 penalty: 1 off
question
factor.
$x^2 - 12x + 32$
answer

Explanation:

Step1: Find two numbers

We need two numbers that multiply to \(32\) (the constant term) and add up to \(-12\) (the coefficient of \(x\)). Let's list the factor pairs of \(32\): \((1, 32)\), \((2, 16)\), \((4, 8)\). Now check the sums: \(1 + 32 = 33\), \(2 + 16 = 18\), \(4 + 8 = 12\). Since the middle term is \(-12x\) and the constant term is positive, both numbers should be negative. So the numbers are \(-4\) and \(-8\) because \((-4)\times(-8)=32\) and \((-4)+(-8)=-12\).

Step2: Factor the trinomial

Using the numbers we found, we can factor \(x^{2}-12x + 32\) as \((x - 4)(x - 8)\) because when we expand \((x - 4)(x - 8)\) using the distributive property (FOIL method), we get \(x^{2}-8x-4x + 32=x^{2}-12x + 32\).

Answer:

\((x - 4)(x - 8)\)