Which expression is a factor of 10×2 + 11x + 3?
A. x − 3
B. 2x + 1
C. 5x − 3
D. 10x + 3
The Correct Answer and Explanation is:
To determine which expression is a factor of the quadratic polynomial (10x^2 + 11x + 3), we can use the factorization method. The goal is to express the quadratic in the form of ((ax + b)(cx + d)), where (a), (b), (c), and (d) are constants.
Step 1: Identify the coefficients
The polynomial is in the standard form (ax^2 + bx + c):
- (a = 10)
- (b = 11)
- (c = 3)
Step 2: Use the AC method
The AC method involves finding two numbers that multiply to (a \cdot c) (the product of the coefficient of (x^2) and the constant term) and add up to (b) (the coefficient of (x)).
Here, (a \cdot c = 10 \cdot 3 = 30) and (b = 11).
We need to find two numbers that multiply to (30) and add to (11):
- The pairs of factors of (30) are: (1 \times 30), (2 \times 15), (3 \times 10), (5 \times 6).
- The pair that adds to (11) is (1) and (10) (since (1 + 10 = 11)).
Step 3: Rewrite the middle term
Using these factors, we rewrite the polynomial as:
[
10x^2 + 10x + x + 3
]
Now, we group the terms:
[
(10x^2 + 10x) + (x + 3)
]
Factoring each group, we get:
[
10x(x + 1) + 1(x + 3)
]
Step 4: Factor by grouping
We can factor out the common binomial:
[
(10x + 1)(x + 3)
]
Step 5: Check the options
Now, let’s check the answer choices:
- A. (x – 3): Not a factor.
- B. (2x + 1): Not a factor.
- C. (5x – 3): Not a factor.
- D. (10x + 3): Part of the factorization, but it needs to be (10x + 1).
Conclusion
None of the provided options are direct factors. However, we can see that the factorization leads us to identify factors of the original polynomial. The complete factorization shows that (10x + 1) and (x + 3) are the factors. To verify, we can also use synthetic division or the quadratic formula, confirming that the expression can be factored, and indeed we notice that (10x^2 + 11x + 3) has no direct matches in the options. Therefore, no answer choice from the given options is correct based on this factorization process.