Question

determine the gcf for each of the polynomials. \\(2x^3 - 7x^2 + 3x\\) the gcf is \\(\boldsymbol{\downarrow}\\) \\(33x^4 - 22\\) the gcf is \\(\boldsymbol{\downarrow}\\) \\(24x^5 - 56x^3 + 16x\\) the gcf is \\(\boldsymbol{\downarrow}\\)

Explanation:

Step1: Find GCF of $2x^3-7x^2+3x$

Step1a: Factor coefficients

Coefficients: 2, -7, 3. GCF of 2,7,3 is 1.

Step1b: Factor variable terms

Variable terms: $x^3, x^2, x$. Lowest power is $x^1=x$.

Step1c: Combine results

GCF = $1 \cdot x = x$

Step2: Find GCF of $33x^4-22$

Step2a: Factor coefficients

Coefficients: 33, -22. GCF of 33,22 is 11.

Step2b: Factor variable terms

Variable terms: $x^4, x^0$ (constant term). Lowest power is $x^0=1$.

Step2c: Combine results

GCF = $11 \cdot 1 = 11$

Step3: Find GCF of $24x^5-56x^3+16x$

Step3a: Factor coefficients

Coefficients: 24, -56, 16. GCF of 24,56,16 is 8.

Step3b: Factor variable terms

Variable terms: $x^5, x^3, x$. Lowest power is $x^1=x$.

Step3c: Combine results

GCF = $8 \cdot x = 8x$

Answer:

For $2x^3 - 7x^2 + 3x$: $x$
For $33x^4 - 22$: $11$
For $24x^5 - 56x^3 + 16x$: $8x$