Question

select all of the factors of (x^3 - 5x^2 + 2x + 8)

(x + 5)
(x - 2)
(x + 4)
(x - 4)
(x + 1)
(x - 1)

Explanation:

🆕 New Concept Discovered: factoring polynomials by grouping
Grouping terms to find common binomial factors.

Step 1: Group the terms

We start with the given cubic polynomial:

$$ x^3 - 5x^2 + 2x - 10 $$

We group the first two terms together and the last two terms together:

$$ (x^3 - 5x^2) + (2x - 10) $$

Step 2: Factor out the greatest common factor (GCF) from each group

From the first group \( (x^3 - 5x^2) \), we factor out the GCF, which is \( x^2 \):

$$ x^2(x - 5) $$

From the second group \( (2x - 10) \), we factor out the GCF, which is \( 2 \):

$$ 2(x - 5) $$

Now we rewrite the expression with these factored groups:

$$ x^2(x - 5) + 2(x - 5) $$

Step 3: Factor out the common binomial

Both terms now share a common binomial factor of \( (x - 5) \). We factor this out:

$$ (x - 5)(x^2 + 2) $$

The factors of the polynomial are \( x - 5 \) and \( x^2 + 2 \). Looking at the given choices, \( x - 5 \) is one of the options.

Answer:

\( x - 5 \)