To factorize the expression \(x^3 - 4x^2 + x + 7\), we can use the technique of "splitting the middle term." This involves breaking up the middle term into two terms such that we can group them in a way that allows us to factor by grouping.
The expression is: \(x^3 - 4x^2 + x + 7\)
Step 1: Group the terms in pairs:
\( (x^3 - 4x^2) + (x + 7) \)
Step 2: Factor out the common terms from each group:
\( x^2 (x - 4) + 1 (x + 7) \)
Now, we have two terms with a common factor in each group.
Step 3: Notice that the terms inside the parentheses are now common. Factor out the common factor:
\( x^2 (x - 4) + 1 (x + 7) \)
\( x^2 (x - 4) + 1 (x - 4) \)
Step 4: Factor out the common factor \((x - 4)\) from both terms:
\( (x - 4)(x^2 + 1) \)
So, the factorization of \(x^3 - 4x^2 + x + 7\) is \((x - 4)(x^2 + 1)\).