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Sagot :
To factor the quadratic expression [tex]\(x^2 + 7x + 12\)[/tex], we need to find two binomials of the form [tex]\((x + a)(x + b)\)[/tex] such that when expanded, they give us the original quadratic expression [tex]\(x^2 + 7x + 12\)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients and constant term:
- The quadratic term's coefficient is [tex]\(1\)[/tex] (for [tex]\(x^2\)[/tex]).
- The linear term's coefficient is [tex]\(7\)[/tex] (for [tex]\(7x\)[/tex]).
- The constant term is [tex]\(12\)[/tex].
2. Find two numbers that multiply to [tex]\(12\)[/tex] and add up to [tex]\(7\)[/tex]:
We need two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
- [tex]\(a \cdot b = 12\)[/tex]
- [tex]\(a + b = 7\)[/tex]
3. List pairs of factors of 12:
The factor pairs of [tex]\(12\)[/tex] are:
- [tex]\(1 \cdot 12\)[/tex]
- [tex]\(2 \cdot 6\)[/tex]
- [tex]\(3 \cdot 4\)[/tex]
- [tex]\(-1 \cdot -12\)[/tex]
- [tex]\(-2 \cdot -6\)[/tex]
- [tex]\(-3 \cdot -4\)[/tex]
4. Determine which pair of factors adds up to [tex]\(7\)[/tex]:
- Checking [tex]\(1\)[/tex] and [tex]\(12\)[/tex]:
- [tex]\(1 + 12 = 13\)[/tex] (Does not match)
- Checking [tex]\(2\)[/tex] and [tex]\(6\)[/tex]:
- [tex]\(2 + 6 = 8\)[/tex] (Does not match)
- Checking [tex]\(3\)[/tex] and [tex]\(4\)[/tex]:
- [tex]\(3 + 4 = 7\)[/tex] (Matches)
- Checking [tex]\(-1\)[/tex] and [tex]\(-12\)[/tex]:
- [tex]\(-1 + (-12) = -13\)[/tex] (Does not match)
- Checking [tex]\(-2\)[/tex] and [tex]\(-6\)[/tex]:
- [tex]\(-2 + (-6) = -8\)[/tex] (Does not match)
- Checking [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex]:
- [tex]\(-3 + (-4) = -7\)[/tex] (Does not match)
5. Construct the binomials:
Since [tex]\(3\)[/tex] and [tex]\(4\)[/tex] are the factors that satisfy both conditions, we have:
[tex]\[ (x + 3)(x + 4) \][/tex]
6. Verify by expanding the binomials:
To ensure our factorization is correct, we can expand [tex]\((x + 3)(x + 4)\)[/tex]:
[tex]\[ (x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12 \][/tex]
The expanded form matches the original quadratic expression.
Therefore, the binomial factors of the expression [tex]\(x^2 + 7x + 12\)[/tex] are:
[tex]\[ \boxed{(x + 4)(x + 3)} \][/tex]
Thus, the correct answer is:
[tex]\[ B. (x+4)(x+3) \][/tex]
### Step-by-Step Solution:
1. Identify the coefficients and constant term:
- The quadratic term's coefficient is [tex]\(1\)[/tex] (for [tex]\(x^2\)[/tex]).
- The linear term's coefficient is [tex]\(7\)[/tex] (for [tex]\(7x\)[/tex]).
- The constant term is [tex]\(12\)[/tex].
2. Find two numbers that multiply to [tex]\(12\)[/tex] and add up to [tex]\(7\)[/tex]:
We need two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that:
- [tex]\(a \cdot b = 12\)[/tex]
- [tex]\(a + b = 7\)[/tex]
3. List pairs of factors of 12:
The factor pairs of [tex]\(12\)[/tex] are:
- [tex]\(1 \cdot 12\)[/tex]
- [tex]\(2 \cdot 6\)[/tex]
- [tex]\(3 \cdot 4\)[/tex]
- [tex]\(-1 \cdot -12\)[/tex]
- [tex]\(-2 \cdot -6\)[/tex]
- [tex]\(-3 \cdot -4\)[/tex]
4. Determine which pair of factors adds up to [tex]\(7\)[/tex]:
- Checking [tex]\(1\)[/tex] and [tex]\(12\)[/tex]:
- [tex]\(1 + 12 = 13\)[/tex] (Does not match)
- Checking [tex]\(2\)[/tex] and [tex]\(6\)[/tex]:
- [tex]\(2 + 6 = 8\)[/tex] (Does not match)
- Checking [tex]\(3\)[/tex] and [tex]\(4\)[/tex]:
- [tex]\(3 + 4 = 7\)[/tex] (Matches)
- Checking [tex]\(-1\)[/tex] and [tex]\(-12\)[/tex]:
- [tex]\(-1 + (-12) = -13\)[/tex] (Does not match)
- Checking [tex]\(-2\)[/tex] and [tex]\(-6\)[/tex]:
- [tex]\(-2 + (-6) = -8\)[/tex] (Does not match)
- Checking [tex]\(-3\)[/tex] and [tex]\(-4\)[/tex]:
- [tex]\(-3 + (-4) = -7\)[/tex] (Does not match)
5. Construct the binomials:
Since [tex]\(3\)[/tex] and [tex]\(4\)[/tex] are the factors that satisfy both conditions, we have:
[tex]\[ (x + 3)(x + 4) \][/tex]
6. Verify by expanding the binomials:
To ensure our factorization is correct, we can expand [tex]\((x + 3)(x + 4)\)[/tex]:
[tex]\[ (x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12 \][/tex]
The expanded form matches the original quadratic expression.
Therefore, the binomial factors of the expression [tex]\(x^2 + 7x + 12\)[/tex] are:
[tex]\[ \boxed{(x + 4)(x + 3)} \][/tex]
Thus, the correct answer is:
[tex]\[ B. (x+4)(x+3) \][/tex]
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