Question

1. simplify the rational expression \frac{9x^4 - 27x^6}{3x^3} for all permissible values of x.

a 3x(1 - 3x)
b 3x(1 - 9x^5)
c 3x - 9x^3
d 9x^3 - 9x^4

Explanation:

Identify the given expression

We are given the rational expression:

$$ \frac{9x^4 - 27x^6}{3x^3} $$

We need to simplify this expression for all permissible values of \(x\) (where \(x
eq 0\)).

Factor the numerator

Find the greatest common factor (GCF) of the terms in the numerator, \(9x^4\) and \(-27x^6\).
The GCF of \(9\) and \(27\) is \(9\).
The GCF of \(x^4\) and \(x^6\) is \(x^4\).
Thus, the GCF of the numerator is \(9x^4\).
Factoring \(9x^4\) out of the numerator gives:

$$ 9x^4 - 27x^6 = 9x^4(1 - 3x^2) $$

Divide by the denominator

Substitute the factored numerator back into the rational expression:

$$ \frac{9x^4(1 - 3x^2)}{3x^3} $$

Divide the coefficients and the variable terms outside the parentheses:

$$ \frac{9x^4}{3x^3} = 3x^{4-3} = 3x $$

Write the simplified expression

Multiply the simplified term by the remaining binomial factor:

$$ 3x(1 - 3x^2) $$

Distribute \(3x\) to check if it matches any other forms:

$$ 3x - 9x^3 $$

Comparing this with the given options:

• Option A: \(3x(1 - 3x)\)
• Option B: \(3x(1 - 9x^5)\)
• Option C: \(3x - 9x^3\)
• Option D: \(9x^3 - 9x^4\) (partially visible)

The simplified expression \(3x - 9x^3\) matches Option C.

Answer:

• A \(3x(1 - 3x)\)
• B \(3x(1 - 9x^5)\)
• C \(3x - 9x^3\) (Correct answer)
• D \(9x^3 - 9x^4\)