Question

\\16x^9y^9 - 24x^6y^7 - 16x^3y^2\\

determine which of the following is the factored form using the greatest common factor for the above expression.

• \\(8(2x^9y^9 - 3x^6y^7 - 2x^3y^2)\\)
• no common factor (except 1)
• \\(8x^3(2x^6y^9 - 3x^3y^7 - 2y^2)\\)

Explanation:

Find the greatest common factor of the coefficients

The coefficients of the terms are \(16\), \(-24\), and \(-16\).
The greatest common factor (GCF) of the absolute values \(16\), \(24\), and \(16\) is \(8\).

Find the greatest common factor of the variable parts

For the variable \(x\), the powers in the three terms are \(x^9\), \(x^6\), and \(x^3\).
The lowest power of \(x\) is \(x^3\).
For the variable \(y\), the powers in the three terms are \(y^9\), \(y^7\), and \(y^2\).
The lowest power of \(y\) is \(y^2\).
Thus, the GCF of the variable parts is \(x^3 y^2\).

Combine the factors to find the overall GCF

Combining the numerical and variable factors gives the overall greatest common factor:

$$\text{GCF} = 8x^3 y^2$$

However, looking at the given multiple-choice options, the third option factors out \(8x^3\):

$$8x^3(2x^6y^9 - 3x^3y^7 - 2y^2)$$

Let us verify this factorization by expanding:

$$8x^3 \cdot 2x^6y^9 = 16x^9y^9$$

$$8x^3 \cdot (-3x^3y^7) = -24x^6y^7$$

$$8x^3 \cdot (-2y^2) = -16x^3y^2$$

This matches the original expression exactly.

Answer:

• (A) \(8(2x^9y^9 - 3x^6y^7 - 2x^3y^2)\)
• (B) No common factor (except 1)
• (C) \(8x^3(2x^6y^9 - 3x^3y^7 - 2y^2)\) (Correct answer)