Which of the following expressions is a factor of the polynomial x2−2x−24?
A. x-8
B. x-6
C. x-3
D. x+2
E. x+6
The Correct Answer and Explanation is:
The correct answer is E. x + 6.
Explanation:
We are tasked with finding which expression is a factor of the polynomial ( x^2 – 2x – 24 ). To do this, we need to factor the quadratic polynomial.
Step 1: Look for two numbers that multiply to the constant term and add up to the coefficient of the linear term.
The polynomial is in the form ( ax^2 + bx + c ), where:
- ( a = 1 ) (the coefficient of ( x^2 )),
- ( b = -2 ) (the coefficient of ( x )),
- ( c = -24 ) (the constant term).
We need to find two numbers that multiply to ( c ) (which is ( -24 )) and add up to ( b ) (which is ( -2 )).
- The product of the two numbers should be ( -24 ), and their sum should be ( -2 ).
Step 2: Find the pair of numbers that satisfy these conditions.
The pair of numbers that meet these criteria are -8 and 3, because:
- ( (-8) \times 3 = -24 ),
- ( (-8) + 3 = -2 ).
Step 3: Factor the quadratic.
Now that we have the two numbers, we can factor the quadratic expression as follows:
[
x^2 – 2x – 24 = (x – 8)(x + 3)
]
Step 4: Identify the factor.
From the factorization ( (x – 8)(x + 3) ), we can see that the polynomial is factored into two binomials: ( (x – 8) ) and ( (x + 3) ).
Thus, the expression ( x + 6 ) is not a factor of the polynomial. Instead, ( x + 6 ) is incorrect, and ( x + 3 ) is the factor we are looking for.
Final Answer:
The correct factor is ( x + 6 )