Question

which function is equivalent to $f(x) = 6x^2 - 23x + 21$?

• $f(x) = (3x - 7)(2x - 3)$
• $f(x) = 3(x - 1)(2x - 7)$
• $f(x) = (6x - 7)(x - 3)$
• $f(x) = 3(x - 7)(2x - 1)$

question 3
which expression is a factor of $x^2 + 7x - 30$?

• $x + 15$

Explanation:

Step1: Expand option A

$$\begin{align*} (3x-7)(2x-3)&=3x\cdot2x + 3x\cdot(-3) + (-7)\cdot2x + (-7)\cdot(-3)\\ &=6x^2 -9x -14x +21\\ &=6x^2 -23x +21 \end{align*}$$

Step2: Verify it matches $f(x)$

The expanded form equals the given $f(x)=6x^2 -23x +21$.

Step3: Factor $x^2+7x-30$

Find two numbers that multiply to $-30$ and add to $7$: $10$ and $-3$.

$$\begin{align*} x^2+7x-30&=x^2+10x-3x-30\\ &=x(x+10)-3(x+10)\\ &=(x+10)(x-3) \end{align*}$$

Step4: Check given option for Q3

The only provided option is $x+15$, which does not match the factors.

Answer:

1. A. $f(x) = (3x-7)(2x - 3)$
2. The provided option $x+15$ is not a factor; the correct factors are $(x+10)$ and $(x-3)$