Q. Factor out the greatest common factor (GCF) from the expression: \(8x^{2} – 12x + 16\).

Factor out the greatest common factor (GCF) from the expression

Given expression: \(8x^{2} – 12x + 16\)

1. List the coefficients and variable parts separately.Coefficients: 8, −12, 16.
   Variable parts: \(x^{2}\), \(x\), and the constant (which is \(x^{0}\)).
2. Find the greatest common divisor of the coefficients.Compute the GCD of 8, 12, and 16. Prime factorizations:

   \(8 = 2^{3}\), \(12 = 2^{2}\cdot 3\), \(16 = 2^{4}\).

   The common prime power is \(2^{2}\), so GCD = 4.

3. Determine the common variable factor.The terms have variable powers \(x^{2}\), \(x^{1}\), and \(x^{0}\). The smallest exponent present is 0, so no positive power of \(x\) is common to all terms. Therefore the GCF does not include \(x\).
4. Write the GCF and divide each term by the GCF.GCF = 4.

   Divide each term by 4:

   \(\dfrac{8x^{2}}{4} = 2x^{2}\), \(\dfrac{-12x}{4} = -3x\), \(\dfrac{16}{4} = 4\).

5. Form the factored expression and verify by distribution.Factored form: \(4\bigl(2x^{2} – 3x + 4\bigr)\).

   Verification by distributing 4:

   \(4\cdot 2x^{2} = 8x^{2}\), \(4\cdot (-3x) = -12x\), \(4\cdot 4 = 16\). These reproduce the original expression.

Final answer: \(4\bigl(2x^{2} – 3x + 4\bigr)\)