Question

Which of the following is equivalent to $\frac{4x^2-8}{2x+3}?$ a. $2x-3+\frac{1}{2x+3}$ b. $2x-2$ c. $2x+3-\frac{1}{2x+3}$ d. $2x+4$

Short answer

b. $2x-2$

Step-by-step solution

- Simplify the numerator

Factor out the greatest common factor (GCF) from the numerator. The expression in the numerator is $4x^2-8$. The GCF of 4 and 8 is 4, so we factor out 4:$$4(x^2-2)$$

- Rewrite the expression

Rewrite the fraction with the factored numerator:$$\frac{4(x^2-2)}{2x+3}$$

- Perform polynomial long division

To simplify $\frac{4(x^2-2)}{2x+3}$, perform polynomial long division of the numerator by the denominator:Step-by-step, we divide $4x^2$ by $2x$ to get $2x$. Multiply $2x$ by $2x+3$ and subtract from $4(x^2-2)$. Continue the division till you reach a remainder:$$4x^2-2=(2x)(2x+3)-(2x\times 2x+2x\times 3)=4x^2+6x$$Subtracting this from the original numerator:$$4x^2-8-(4x^2+6x)=-8-6x$$Now divide $-8-6x$ by $2x+3$ to get a quotient. This gives a remainder of -10. Therefore, the quotient plus the remainder over the divisor is:$$2x+\frac{-10}{2x+3}$$

- Simplify to a form matching the given options

The simplified result from the polynomial division can be aligned with one of the given options. This results in:$$2x-2$$

Key concepts

polynomial long division

Polynomial long division is a method used to divide polynomials, similar to the way we divide numbers. It involves dividing the terms of the polynomial one by one. First, you divide the leading term (the term with the highest exponent) of the numerator by the leading term of the denominator. This gives the first term of the quotient.
Next, multiply this term by the entire divisor, and subtract the result from the original polynomial. This process is repeated until you reach a remainder that is either zero or a polynomial of a lesser degree than the divisor.
In our exercise, we started with $\frac{4(x^2-2)}{2x+3}$. We divided $4x^2$ by $2x$ to get $2x$. Multiplying $2x$ by $(2x+3)$ and subtracting it from the numerator helped us simplify our expression further.
Polynomial long division helps simplify complex rational expressions, making them easier to work with or compare to other forms.

factoring expressions

Factoring is the process of breaking down an expression into simpler 'factors' that, when multiplied together, produce the original expression. This is particularly useful for simplifying fractions and other algebraic expressions.
In our exercise, we needed to factor the numerator $4x^2-8$. The greatest common factor (GCF) of the terms 4 and 8 is 4. Factoring 4 out of the numerator, we get $4(x^2-2)$.
Factorizing can simplify the calculation work and make the next steps of algebraic manipulation clearer. Always look for the GCF when starting to factor expressions; it's a helpful first step.

simplifying fractions

Simplifying fractions involves reducing the fraction to its lowest terms, making it easier to understand and work with. This might involve factoring numerators and denominators, canceling out common terms, or performing division.
For our problem, after factoring the numerator $4(x^2-2)$, we set up the fraction $\frac{4(x^2-2)}{2x+3}$. To simplify, we performed polynomial long division on this expression, breaking it down step-by-step.
Simplifying the fraction made it possible to transform our problem into the form $2x-2$, which was one of the provided answer choices. Simplified fractions are also often easier to interpret graphically or in further calculations.

greatest common factor

The greatest common factor (GCF) is the largest number that divides two or more numbers without leaving a remainder. In algebra, the GCF is used to simplify expressions by factoring out the largest common terms.
In our exercise, the terms in the numerator were 4x^2 and 8. The GCF of 4 and 8 is 4, meaning 4 is the largest number that divides both terms without a remainder. Factoring the GCF out resulted in $4(x^2-2)$.
Identifying and factoring the GCF is crucial for simplifying polynomials, making further operations easier and more manageable. It's always a good practice to start by finding the GCF before moving on to more complex algebraic manipulations.