To factorize the polynomial [tex]\( -x^2 - 15x - 54 \)[/tex], follow these steps:
1. Identify the polynomial structure:
The polynomial given is in the form [tex]\( -x^2 - 15x - 54 \)[/tex].
2. Consider the constant factor:
Notice that there is a negative sign in front of the polynomial. This means we can factor out [tex]\(-1\)[/tex] first to simplify the expression.
[tex]\[
-x^2 - 15x - 54 = -1(x^2 + 15x + 54)
\][/tex]
3. Factor the quadratic polynomial inside the parentheses:
Next, we need to factor the quadratic polynomial [tex]\( x^2 + 15x + 54 \)[/tex].
4. Find two numbers that multiply to 54 and add up to 15:
The constant term is 54, and the coefficient of the linear term is 15. We need two numbers that multiply to 54 and add to 15. These numbers are 6 and 9 because:
[tex]\[
6 \times 9 = 54 \quad \text{and} \quad 6 + 9 = 15
\][/tex]
5. Write the factors of the quadratic polynomial:
Given the numbers found, the quadratic polynomial [tex]\( x^2 + 15x + 54 \)[/tex] can be factored as:
[tex]\[
(x + 6)(x + 9)
\][/tex]
6. Combine with the extracted factor [tex]\(-1\)[/tex]:
Now, we put the [tex]\(-1\)[/tex] we factored out back in:
[tex]\[
-1(x + 6)(x + 9)
\][/tex]
Thus, the factorization of the polynomial [tex]\( -x^2 - 15x - 54 \)[/tex] is:
[tex]\[
-(x + 6)(x + 9)
\][/tex]
The correct answer is:
B. [tex]\( -1(x + 9)(x + 6) \)[/tex]
