Question

Divide \(\frac{21 x^3-14 x^2}{14 x^3+21 x^2}\) A. \(\frac{3 x-2}{2 x+3}\) B. \(\frac{21 x-14}{14 x+12}\) C. \(\frac{3 x^2-2 x}{2 x^2+3 x}\) D. \(\frac{21 x^2-14 x}{14 x^2+12 x}\)

Short answer

A. \ \( \frac{3 x - 2}{2 x + 3} \)

Step-by-step solution

Identify the expression

The given expression to be simplified is \ \( \frac{21 x^3 - 14 x^2}{14 x^3 + 21 x^2} \).

Factor out the common terms

Factor out the greatest common divisor in both the numerator and the denominator. For the numerator: \ \( 21 x^3 - 14 x^2 = 7 x^2 (3x - 2) \). For the denominator: \ \( 14 x^3 + 21 x^2 = 7 x^2 (2x + 3) \).

Simplify the fraction

Substitute the factored forms back into the fraction: \ \( \frac{7 x^2 (3x - 2)}{7 x^2 (2x + 3)} \). Cancel out the common factor \ \( 7 x^2 \). The simplified fraction is \ \( \frac{3x - 2}{2x + 3} \).

Match with the given choices

Now match the simplified fraction to the given choices. The correct choice is A: \ \( \frac{3 x - 2}{2 x + 3} \).

Key concepts

Factoring Polynomials

Factoring polynomials is like breaking down a number into its prime factors. Instead of prime numbers, we break a polynomial into simpler polynomials called 'factors'.
To factor a polynomial, look for the greatest common factors among the terms. For example, in the polynomial 21x^3 - 14x^2, notice that both terms share the factor 7x^2. So, you can rewrite it as 7x^2(3x - 2).

Similarly, for the polynomial 14x^3 + 21x^2, factor out 7x^2 to get 7x^2(2x + 3).
Factoring helps in simplifying complex rational expressions, making them easier to divide or simplify further.

Greatest Common Divisor

The greatest common divisor (GCD) is crucial for simplifying polynomials. It's the largest factor that divides all terms of a polynomial without leaving a remainder.

When simplifying the expression \( \frac{21x^3 - 14x^2}{14x^3 + 21x^2} \), we found the GCD in both the numerator and the denominator: 7x^2.
By dividing both parts by their GCD, we get \( \frac{7x^2(3x - 2)}{7x^2(2x + 3)} \). This step is essential for simplifying the expression and making the terms easier to work with.
Cancelling out the GCD simplifies the fraction to \( \frac{3x - 2}{2x + 3} \), optimizing the simplification process.

Algebraic Simplification

Algebraic simplification involves making expressions shorter and easier to handle. It's like cleaning up a messy room by putting things back into their rightful places.

Starting with the given expression \( \frac{21x^3 - 14x^2}{14x^3 + 21x^2} \), simplify by factoring out the greatest common divisor. This results in \( \frac{7x^2 (3x - 2)}{7x^2 (2x + 3)} \).

Next, cancel out the common terms (7x^2 in this case), leaving us with \( \frac{3x - 2}{2x + 3} \). This is the simplest form of the expression, which is much easier to interpret and use.
Simplifying rational expressions this way is fundamental in algebra, helping solve more complex equations or performing operations like addition, subtraction, multiplication, and division.