Question

1 let f(x) = 14x³ + 28x² - 46x and g(x) = 2x + 7. which is the solution set to the equation (1/12)f(x) = g(x)? a {-3, 0, 1} c {-2, 1, 3} b {-3, -1, 2} d {1, 5, 11} options: c, a, b, d

Explanation:

Step1: Substitute \( f(x) \) into the equation

Given \( \frac{1}{12}f(x)=g(x) \), substitute \( f(x) = 14x^3 + 28x^2 - 46x \) and \( g(x)=2x + 7 \):

$$ \frac{1}{12}(14x^3 + 28x^2 - 46x)=2x + 7 $$

Multiply both sides by 12 to eliminate the denominator:

$$ 14x^3 + 28x^2 - 46x = 24x + 84 $$

Step2: Rearrange into standard polynomial form

Subtract \( 24x + 84 \) from both sides:

$$ 14x^3 + 28x^2 - 70x - 84 = 0 $$

Divide all terms by 14 to simplify:

$$ x^3 + 2x^2 - 5x - 6 = 0 $$

Step3: Factor the polynomial

Try rational roots. Possible roots are \( \pm1, \pm2, \pm3, \pm6 \).
Test \( x = -1 \): \( (-1)^3 + 2(-1)^2 - 5(-1) - 6 = -1 + 2 + 5 - 6 = 0 \), so \( (x + 1) \) is a factor.
Divide \( x^3 + 2x^2 - 5x - 6 \) by \( (x + 1) \) using polynomial long division or synthetic division.
Synthetic division with root -1:

$$
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$$

So the polynomial factors as \( (x + 1)(x^2 + x - 6) \).
Factor \( x^2 + x - 6 \): \( (x + 3)(x - 2) \).
Thus, the factored form is \( (x + 1)(x + 3)(x - 2) = 0 \).

Step4: Solve for \( x \)

Set each factor to zero:

• \( x + 1 = 0 \Rightarrow x = -1 \)
• \( x + 3 = 0 \Rightarrow x = -3 \)
• \( x - 2 = 0 \Rightarrow x = 2 \)

The solution set is \( \{-3, -1, 2\} \), which matches option B.

Answer:

B. \(\{-3, -1, 2\}\)