The factors of `x^2-7x+6` are

Options

A. `x(x-6)(x-1)`
B. `(x^2-6)(x-1)`
C. `(x+1)(x+2)(x-3)`
D. `(x-1)(x+3)\ (x-2)`

Answer

To factor the expression \(x^2 - 7x + 6\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = -7\) (coefficient of \(x\))
- \(c = 6\) (constant term)

### Step 2: Calculate the product \(a \cdot c\)
We need to find the product \(a \cdot c\):
\[ a \cdot c = 1 \cdot 6 = 6 \]

### Step 3: Find two numbers that multiply to \(a \cdot c\) and add up to \(b\)
We need to find two numbers that multiply to \(6\) (the product from Step 2) and add up to \(-7\) (the value of \(b\)). The numbers that satisfy these conditions are:
\[ -6 \text{ and } -1 \]
because:
\[ -6 \cdot -1 = 6 \quad \text{and} \quad -6 + (-1) = -7 \]

### Step 4: Rewrite the middle term using the two numbers
We can rewrite the expression \(x^2 - 7x + 6\) by splitting the middle term:
\[ x^2 - 6x - 1x + 6 \]

### Step 5: Factor by grouping
Now, we can group the terms:
\[ (x^2 - 6x) + (-1x + 6) \]
Next, we factor out the common factors from each group:
\[ x(x - 6) - 1(x - 6) \]

### Step 6: Factor out the common binomial
Now, we can factor out the common binomial \((x - 6)\):
\[ (x - 6)(x - 1) \]

### Final Answer
Thus, the factors of the expression \(x^2 - 7x + 6\) are:
\[ (x - 1)(x - 6) \]

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