To factor the polynomial [tex]\( x^2 - 4x - 12 \)[/tex], we need to determine the missing term such that it completes the factorization. Here is the step-by-step solution to find the missing term:
1. Identify the polynomial: We are given the polynomial [tex]\( x^2 - 4x - 12 \)[/tex].
2. Factoring the polynomial: To factor the polynomial, we need to find two numbers that multiply to the constant term (-12) and add up to the coefficient of the linear term (-4).
3. Setting up a pair of equations:
- Let [tex]\( a \)[/tex] and [tex]\( b \)[/tex] be the numbers we are looking for.
- [tex]\( a \cdot b = -12 \)[/tex] (They multiply to -12).
- [tex]\( a + b = -4 \)[/tex] (They add up to -4).
4. Identifying the numbers:
- After considering pairs of factors of -12, we find that [tex]\( a = -6 \)[/tex] and [tex]\( b = 2 \)[/tex] fit both equations:
- [tex]\(-6 \cdot 2 = -12\)[/tex]
- [tex]\(-6 + 2 = -4\)[/tex]
5. Writing the factorization: Now we can rewrite the polynomial [tex]\( x^2 - 4x - 12 \)[/tex] in its factored form:
- [tex]\( x^2 - 4x - 12 = (x - 6)(x + 2) \)[/tex]
6. Substituting the missing term: Given the form [tex]\( (x - 6)(x + \underline{\ \ }) \)[/tex], we see that the missing term is [tex]\( 2 \)[/tex].
Thus, the missing term that Simon needs to complete the factorization is:
[tex]\(\boxed{2}\)[/tex]
