Question

Which of the following is equivalent to \(\frac{z^2+7 z-3}{z+2}\) ? A) \(z+5-\frac{13}{z+2}\) B) \(z+5-\frac{7}{z+2}\) C) \(z+9-\frac{21}{z-2}\) D) \(z+9-\frac{15}{z-2}\)

Short answer

The equivalent form of the given rational expression \(\frac{z^2+7z-3}{z+2}\) is \(\mathbf{z+5-\frac{13}{z+2}}\).

Step-by-step solution

Setup Polynomial Long Division

We will perform polynomial long division of the given expression: \(\frac{z^2+7z-3}{z+2}\). We will divide \(z^2+7z-3\) by \(z+2\) and try to obtain a result that matches one of the given choices.

Perform Polynomial Long Division

Divide \(z^2+7z-3\) by \(z+2\): 1. Divide \(\frac{z^2}{z}\) and write down the result: \(z\). 2. Multiply \((z+2)\) by \(z\) and subtract the result from the dividend: \((z^2+7z-3) - (z^2+2z) = 5z-3\). 3. Divide \(\frac{5z}{z}\) and write down the result: \(+5\). 4. Multiply \((z+2)\) by \(5\) and subtract the result from the current dividend: \((5z-3) - (5z+10) = -13\). 5. Since the degree of the remaining dividend \(-13\) is less than the degree of the divisor \((z+2)\), we stop. Thus, the result of our polynomial long division is \(z+5-\frac{13}{z+2}\).

Compare Result with Choices

We obtained the result \(z+5-\frac{13}{z+2}\). Now let's compare it with the given choices: A) \(z+5-\frac{13}{z+2} \) (Match!) B) \(z+5-\frac{7}{z+2}\) C) \(z+9-\frac{21}{z-2}\) D) \(z+9-\frac{15}{z-2}\)

Solution

Since our result matches choice (A), the equivalent form of the given rational expression \(\frac{z^2+7z-3}{z+2}\) is \(\mathbf{z+5-\frac{13}{z+2}}\).

Key concepts

Rational Expressions

Rational expressions are like fractions but, instead of numbers, we have polynomials in the numerator and denominator. It's crucial to note that the denominator cannot be zero, as division by zero is undefined. In our exercise, the rational expression is \( \frac{z^2+7z-3}{z+2} \).

• The numerator is a polynomial \( z^2 + 7z - 3 \).
• The denominator is another polynomial \( z + 2 \).

Understanding rational expressions helps manage complex algebraic fractions. They are common in algebra problems and are often tested in standardized tests like the SAT. The goal is to simplify or transform these expressions to make them easier to handle and understand.
In our exercise, we transform it using polynomial long division to find its equivalent expression.

Polynomials

Polynomials are expressions made up of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, \( z^2 + 7z - 3 \) is a polynomial of degree 2.

• The degree of a polynomial is the highest power of the variable in the expression.
• Understanding polynomial degree is crucial because it determines the behavior of polynomial division.

When dividing polynomials, the highest degree terms are divided first. Identifying and manipulating each part of the polynomial is key to successful division and simplification. This skill helps you to understand and solve diverse mathematical problems from simple algebra to complex calculus.

Algebraic Division

Algebraic division, much like long division with numbers, involves dividing polynomials. This process helps simplify expressions and solve polynomial equations. Let's break down the steps used in the division from our example:

• **Divide**: Start with the highest degree term in the dividend and divide it by the highest degree term in the divisor. In our case, divide \( z^2 \) by \( z \) to get \( z \).
• **Multiply**: Multiply the entire divisor by \( z \) and subtract the result from the dividend.
• **Repeat**: Continue the process with the remainder. Divide the new leading term by the leading term of the divisor.
• **Stop**: When no further division is possible (when the remainder is of lesser degree than the divisor), you're finished!

This process gives us a quotient, which is the simplified form of our original expression, plus a remainder. Algebraic division is important because it splits complex polynomial structures into more manageable parts.

SAT Math Preparation

Preparing for the SAT involves understanding a wide range of mathematical concepts, including algebra and the ability to manipulate expressions and equations with tools like polynomial long division. Mastery of solving rational expressions helps:

• Boost confidence in tackling algebra problems.
• Enhance mathematical agility in solving and simplifying complex expressions.

Polynomial division techniques play a role in this preparation. Gaining familiarity with breaking down expressions, manipulating terms, and simplifying complex equations prepares you well for test day. A strong grasp of these concepts not only leads to better SAT scores but also lays the foundation for future mathematical learning.