To factor the given polynomial \( x^2 - 11x - 26 \) over the integers, we need to express it as a product of two binomials. Let's label the polynomial as \( P(x) \):
[tex]\[ P(x) = x^2 - 11x - 26 \][/tex]
To factor this, we look for two numbers that multiply to give the constant term, \(-26\), and add up to the coefficient of the linear term, \(-11\).
The expression can be factored in the following steps:
### Step 1: Identify the constant term and the coefficient of the linear term
- The constant term is \(-26\).
- The coefficient of the \( x \) term is \(-11\).
### Step 2: Find two numbers that multiply to \(-26\) and have a sum of \(-11\)
- These numbers are \( -13 \) and \( 2 \):
- \( -13 \times 2 = -26 \)
- \( -13 + 2 = -11 \)
### Step 3: Write the polynomial as a product of two binomials
With these numbers, the quadratic polynomial can be factored as follows:
[tex]\[ x^2 - 11x - 26 = (x - 13)(x + 2) \][/tex]
Thus, the factored form of the polynomial \( x^2 - 11x - 26 \) is:
[tex]\[ \boxed{(x - 13)(x + 2)} \][/tex]
