Which expression is the factorization of x^2 + 10x + 21? O(x + 3)(x + 7) (x + 4)(x + 6) â—‹ (x + 6)(x + 15) O (x + 7)(x + 14)
The Correct Answer and Explanation is:
The correct factorization of the expression x² + 10x + 21 is:
✔ (x + 3)(x + 7)
Explanation:
To factor a quadratic expression of the form x² + bx + c, we look for two numbers that:
- Multiply to give the constant term c, which is 21
- Add to give the middle coefficient b, which is 10
Let’s identify all the factor pairs of 21:
- 1 and 21 → 1 + 21 = 22
- 3 and 7 → 3 + 7 = 10
- Other combinations like -3 and -7 also multiply to 21, but their sum is -10
So, the pair 3 and 7 works because:
- 3 × 7 = 21 (gives the constant)
- 3 + 7 = 10 (gives the middle term)
This means we can write the expression as:
x² + 10x + 21 = (x + 3)(x + 7)
Why the other choices are incorrect:
- (x + 4)(x + 6):
4 × 6 = 24, and 4 + 6 = 10 → sum matches but product is 24, not 21 - (x + 6)(x + 15):
6 × 15 = 90, and 6 + 15 = 21 → product is wrong, too big - (x + 7)(x + 14):
7 × 14 = 98, and 7 + 14 = 21 → sum matches the constant, not the middle term
Final Answer:
(x + 3)(x + 7) is the correct factorization of x² + 10x + 21.
