Question
Factor \( x^{2}+5 x+6 \) \( (x+5)(x+1) \) \( (x-5)(x-1) \) \( (x+3)(x+2) \) \( (x-3)(x-2) \)
Ask by Schofield Simmons. in the United States
Mar 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The polynomial \( x^{2}+5x+6 \) factors to \( (x+3)(x+2) \).
Solution
We begin with the quadratic polynomial
\[
x^2 + 5x + 6.
\]
We look for two numbers \(a\) and \(b\) such that
\[
a + b = 5 \quad \text{and} \quad a \cdot b = 6.
\]
Considering the factors of \(6\), we find:
\[
2 + 3 = 5 \quad \text{and} \quad 2 \times 3 = 6.
\]
Thus, the polynomial factors as
\[
(x+2)(x+3).
\]
Since multiplication is commutative, this is equivalent to
\[
(x+3)(x+2).
\]
The correct answer is \((x+3)(x+2)\).
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Simplify this solution The Deep Dive
To factor the quadratic \( x^{2}+5x+6 \), we need to find two numbers that add up to 5 (the coefficient of \( x \)) and multiply to 6 (the constant term). The numbers 2 and 3 satisfy these conditions, leading us to the factorization \( (x+2)(x+3) \). Is it just me, or does factorization feel like dressing up numbers in their finest outfits? Every quadratic has a unique style, and when you find the right factors, it's like matching the perfect tie with a crisp shirt! So, next time you're working on a quadratic, keep an eye out for those number pairs that help them shine through!
