Sure, let's go through the steps to factor the polynomial [tex]\(x^2 - 11x + 28\)[/tex].
1. Understand the Polynomial: The polynomial we have is [tex]\(x^2 - 11x + 28\)[/tex].
2. Factoring the Polynomial: We want to find two numbers that multiply together to give the constant term (28) and add together to give the coefficient of the linear term (-11).
3. Finding the Factors:
- The constant term is 28.
- We need to find two numbers that multiply to 28 and sum to -11. The numbers -7 and -4 satisfy this condition because:
- (-7) * (-4) = 28
- (-7) + (-4) = -11
4. Writing the Factors: Hence, we can factor [tex]\(x^2 - 11x + 28\)[/tex] as:
[tex]\[
(x - 7)(x - 4)
\][/tex]
So, the factored form of the polynomial [tex]\(x^2 - 11x + 28\)[/tex] is:
[tex]\[
(x - 7)(x - 4)
\][/tex]
Now, putting this into the form suggested by the question:
[tex]\[
(x + \square)(x - 7)
\][/tex]
Since we know:
[tex]\[
(x - 7)(x - 4)
\][/tex]
Here, the correct number to fill in the blank would be -4.
Thus, the expression can be re-written as:
[tex]\[
(x - 4)(x - 7)
\][/tex]
These are the steps to factor the polynomial and re-write it as per the model suggested in the question.
