Question

practice factoring polynomial expressions.
choose which polynomials have a greatest common factor of $3x$.
mark all correct answers
\\(\square\\) a. \\(x^2 + 6\\)
\\(\square\\) b. \\(3x^2 + 9x\\)
\\(\square\\) c. \\(9x^4 + 5x + 1\\)
\\(\square\\) d. \\(9x^4 + 6x^3 + 3x\\)

Explanation:

Step1: Check polynomial a

Factor terms of $x^2+6$: terms are $x^2$ (GCF coefficient 1, variable $x^2$) and $6$ (GCF coefficient 6, no variable). Overall GCF is $1$, not $3x$.

Step2: Check polynomial b

Factor $3x^2+9x$:

$$\begin{align} 3x^2+9x &= 3x(x) + 3x(3) \\ &= 3x(x+3) \end{align}$$

GCF is $3x$.

Step3: Check polynomial c

Factor terms of $9x^4+5x+1$: terms are $9x^4$ (coefficient 9, $x^4$), $5x$ (coefficient 5, $x$), $1$ (coefficient 1, no variable). GCF of coefficients is 1, so overall GCF is $1$, not $3x$.

Step4: Check polynomial d

Factor $9x^4+6x^3+3x$:

$$\begin{align} 9x^4+6x^3+3x &= 3x(3x^3) + 3x(2x^2) + 3x(1) \\ &= 3x(3x^3+2x^2+1) \end{align}$$

GCF is $3x$.

Answer:

b. $3x^2 + 9x$
d. $9x^4 + 6x^3 + 3x$