Question

factor by grouping.

\\x^3 - 4x^2 + 5x - 20\\

\\x^3 - 4x^2 + 5x - 20 = (x - 4)(x^2 + 5)\\

Explanation:

🆕 New Concept Discovered: Factoring by Grouping
Pairing terms to find common binomial factors.

Step 1: Group the terms

Group the four-term polynomial into two pairs:

$$ (x^3 - 4x^2) + (5x - 20) $$

Step 2: Factor out the GCF from each group

Find the greatest common factor (GCF) for each pair:

• For the first group, \( x^3 - 4x^2 \), the GCF is \( x^2 \):

$$ x^2(x - 4) $$

• For the second group, \( 5x - 20 \), the GCF is \( 5 \):

$$ 5(x - 4) $$

Combine these factored groups:

$$ x^2(x - 4) + 5(x - 4) $$

Step 3: Factor out the common binomial

Both terms now share a common binomial factor of \( (x - 4) \). Factor this out to get the final expression:

$$ (x - 4)(x^2 + 5) $$

Answer:

$$ (x - 4)(x^2 + 5) $$