Problem 44 - Algebra 1: Concepts and Skills

Question

Write the quotient in simplest form. $$\frac{3 x^{2}+2 x-8}{3 x} \div(3 x-4)$$

Step-by-Step Solution

Answer

The simplified quotient is \( \frac{x+2}{3x} \).

Step 1: Identify the Dividing Fraction

The problem can also be written as \( \frac{3 x^{2}+2 x-8}{3 x} \) * \( \frac{1}{3 x-4} \) because division by a fraction is equivalent to multiplication by its reciprocal.

Step 2: Factor the Polynomial

Factor the polynomials in the numerator of the first fraction to simplify further, it factors into \( 3x^{2}+2x-8 = (3x-4)(x+2) \). So the equation now becomes \( \frac{(3x-4)(x+2)}{3x} \) * \( \frac{1}{3x-4} \).

Step 3: Simplify the Fractions

The \( 3x-4 \) terms in the numerator of the first fraction and the denominator of the second fraction cancel out, resulting in \( \frac{x+2}{3x} \).

Step 4: Final Simplification

The final result is \( \frac{x+2}{3x} \) which is the simplified form of the original expression.

Key Concepts

Polynomial Division

Polynomial division is an essential skill in algebra when working with rational expressions. In this context, dividing polynomials means finding a quotient when one polynomial is divided by another. It's similar to long division with numbers but involves variables. Here, you can either use long division or synthetic division (when applicable) depending on the divisor's degree.

• When dividing, rewrite the expression as a multiplication problem by using the reciprocal of the divisor.
• Identify the numerator and divisor to transform the division into multiplication by flipping the divisor and changing the division sign to multiplication.
• This problem is a fraction division, which can be simplified by multiplying by the reciprocal.

For example, given the expression \( \frac{3x^2 + 2x - 8}{3x} \div (3x - 4) \), rearrange it to \( \frac{3x^2 + 2x - 8}{3x} \times \frac{1}{3x - 4} \). This format allows easier simplification using factoring and cancellations.

Factoring Polynomials

Factoring polynomials is breaking down complex expressions into products of simpler polynomials. It's crucial when simplifying algebraic expressions, especially in polynomial division. To factor a polynomial, look for common factors or apply techniques like grouping, using special formulas, or employing the quadratic formula.

• Start by finding factors of polynomial terms that can be simplified or canceled. This makes the entire expression less complex.
• For quadratic expressions \( ax^2 + bx + c \), look for two numbers that multiply to \( ac \) and add up to \( b \).
• In the example, \( 3x^2 + 2x - 8 \) factors into \( (3x-4)(x+2) \).

By identifying this factorization, the rational expression simplifies significantly when terms can be canceled out later.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. It's crucial for clarity and ease of calculation. In rational expressions, this often means canceling identical factors from the numerator and denominator.

• Once polynomials are factored, look for common terms across the numerator and the denominator that can be canceled out.
• Ensure that the factors you cancel are indeed present in both the numerator and denominator.
• Simplification often involves recognizing the greatest common factor (GCF) and reducing the expression step by step.

In our initial expression, after factoring and rearranging, notice \( (3x-4) \) appears in both the numerator and the denominator, allowing it to be canceled, yielding \( \frac{x+2}{3x} \) as the simplest form. This process verifies that no further reduction is possible and confirms the solution.