Question

consider the polynomial $6x^2 - 8x + 2$. what is the gcf of the polynomial? what is the equivalent factored form of the polynomial?

Explanation:

Step1: Find GCF of coefficients

Coefficients: 6, -8, 2. The greatest common factor is 2.

Step2: Find GCF of variable terms

Variable terms: $x^2$, $x$, $x^0$. The lowest power of $x$ is $x^0=1$, so GCF for variables is 1.

Step3: Combine GCFs

Overall GCF = $2 \times 1 = 2$

Step4: Factor out GCF from polynomial

$6x^2 - 8x + 2 = 2(3x^2 - 4x + 1)$

Step5: Factor quadratic inside parentheses

Factor $3x^2 - 4x + 1$: find two numbers that multiply to $3 \times 1 = 3$ and add to -4. The numbers are -1 and -3.
$3x^2 - 3x - x + 1 = 3x(x-1) -1(x-1) = (3x-1)(x-1)$

Step6: Write fully factored form

Substitute back: $2(3x-1)(x-1)$

Answer:

GCF: $2$
Equivalent factored form: $2(3x-1)(x-1)$