Question

1. $4b + 3b^2 - 10b^4$
2. $20v^2 - 4v + 8$
3. $6m^4n^2 + 9mn$

Explanation:

Problem 6: \( 4b + 3b^2 - 10b^4 \)

Step 1: Identify the GCF

The terms are \( 4b \), \( 3b^2 \), and \( -10b^4 \). The GCF of the coefficients \( 4 \), \( 3 \), \( -10 \) is \( 1 \), and the GCF of the variables is \( b \) (since the lowest power of \( b \) is \( 1 \)).

Step 2: Factor out the GCF

Factor out \( b \) from each term:
\( 4b + 3b^2 - 10b^4 = b(4 + 3b - 10b^3) \)
We can also rearrange the polynomial in standard form (descending powers of \( b \)) first: \( -10b^4 + 3b^2 + 4b \), and then factor out \( -b \) (to make the leading coefficient positive):
\( -10b^4 + 3b^2 + 4b = -b(10b^3 - 3b - 4) \) (optional, depending on preference)

Step 1: Identify the GCF

The coefficients are \( 20 \), \( -4 \), \( 8 \). The GCF of \( 20 \), \( 4 \), \( 8 \) is \( 4 \). There is no variable common to all terms (only \( v^2 \), \( v \), and a constant term \( 8 \)), so the GCF is \( 4 \).

Step 2: Factor out the GCF

Factor out \( 4 \) from each term:
\( 20v^2 - 4v + 8 = 4(5v^2 - v + 2) \)

Step 1: Identify the GCF

For the coefficients \( 6 \) and \( 9 \), the GCF is \( 3 \). For the variables, the lowest power of \( m \) is \( 1 \) and the lowest power of \( n \) is \( 1 \), so the GCF of the variables is \( mn \). Combining these, the GCF of the two terms is \( 3mn \).

Step 2: Factor out the GCF

Factor out \( 3mn \) from each term:
\( 6m^4n^2 + 9mn = 3mn(2m^3n + 3) \)

Answer:

\( b(4 + 3b - 10b^3) \) (or \( -b(10b^3 - 3b - 4) \))

Problem 8: \( 20v^2 - 4v + 8 \)