Question

factor completely.

\\8r^3 - 20r^2 + 6r - 15\\

Explanation:

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

Step 1: Group the terms

When we have a four-term polynomial, we can group them into two pairs to look for common factors in each pair:

$$ (8r^3 - 20r^2) + (6r - 15) $$

Step 2: Factor out the GCF from each group

For the first group, \( 8r^3 - 20r^2 \), the greatest common factor (GCF) is \( 4r^2 \):

$$ 4r^2(2r - 5) $$

For the second group, \( 6r - 15 \), the GCF is \( 3 \):

$$ 3(2r - 5) $$

Now, write the grouped expression:

$$ 4r^2(2r - 5) + 3(2r - 5) $$

Step 3: Factor out the common binomial

Both terms now share a common binomial factor of \( (2r - 5) \). We can pull this common factor out to complete the factoring:

$$ (2r - 5)(4r^2 + 3) $$

Answer:

$$ (2r - 5)(4r^2 + 3) $$