Question

1. choose at least 2 of the following. fully common factor the expressions. show your work.

a) \\(8x - 12\\)

b) \\(-4x^2 + 6\\)

c) \\(-14x^2 + 16x\\)

Explanation:

🆕 New Concept Discovered: Greatest Common Factor (GCF) Factoring
Finding the largest shared factor of all terms.

Step 1: Factor the first expression \( 8x - 12 \)

To fully common factor the expression, find the Greatest Common Factor (GCF) of the terms \( 8x \) and \( -12 \).

• The GCF of the coefficients \( 8 \) and \( 12 \) is \( 4 \).
• Divide each term by \( 4 \) and write the factored expression:

$$ 8x - 12 = 4(2x - 3) $$

Step 2: Factor the second expression \( -4x^2 + 6 \)

Find the GCF of the terms \( -4x^2 \) and \( 6 \).

• The GCF of the coefficients \( 4 \) and \( 6 \) is \( 2 \).
• When the leading term is negative, it is standard practice to factor out the negative sign along with the GCF, which is \( -2 \).
• Divide each term by \( -2 \):

$$ -4x^2 + 6 = -2(2x^2 - 3) $$

Step 3: Factor the third expression \( -14x^2 + 16x \)

Find the GCF of the terms \( -14x^2 \) and \( 16x \).

• The GCF of the coefficients \( 14 \) and \( 16 \) is \( 2 \).
• Both terms share the variable \( x \), so the GCF of the variable parts is \( x \).
• Factoring out the negative leading coefficient gives a GCF of \( -2x \).
• Divide each term by \( -2x \):

$$ -14x^2 + 16x = -2x(7x - 8) $$

(Note: If you factor out positive \( 2x \) instead, the expression is \( 2x(-7x + 8) \).)

Answer:

Here are the fully factored forms for all three expressions:

1. \( 8x - 12 \)

$$ 4(2x - 3) $$

1. \( -4x^2 + 6 \)

$$ -2(2x^2 - 3) $$

1. \( -14x^2 + 16x \)

$$ -2x(7x - 8) \quad \text{or} \quad 2x(-7x + 8) $$