Question

\(y=\frac{3 x^{2}+7}{x-3}\) Which of the following expressions is equivalent to \(y ?\) a. \(3 x+9-\frac{20}{x-3}\) b. \(3 x+9+\frac{34}{x-3}\) c. \(3 x+43\) d. \(3 x^{2}+\frac{9}{x-3}\)

Short answer

b. \(3x + 9 + \frac{34}{x - 3}\).

Step-by-step solution

Perform polynomial long division

Perform polynomial long division to divide the numerator by the denominator. We are dividing: \( \frac{3x^2 + 7}{x - 3} \).First, divide the leading term of the numerator by the leading term of the denominator: \( \frac{3x^2}{x} = 3x \).

Multiply and subtract

Multiply the entire divisor (\( x - 3 \)) by \( 3x \), resulting in:\( 3x(x - 3) = 3x^2 - 9x \).Subtract this result from the original polynomial: \( (3x^2 + 7) - (3x^2 - 9x) = 9x + 7 \).

Repeat the division process

Take the new polynomial (\( 9x + 7 \)) and divide the leading term by the leading term of the divisor: \( \frac{9x}{x} = 9 \).Multiply the entire divisor (\( x - 3 \)) by \( 9 \), resulting in:\( 9(x - 3) = 9x - 27 \).Subtract this result from the new polynomial: \( (9x + 7) - (9x - 27) = 34 \).

Write out the complete quotient and remainder

The complete quotient from the division process is the sum of the terms obtained in each division step:\( y = 3x + 9 + \frac{34}{x - 3} \).

Identify the equivalent expression

Compare the obtained expression to the given options:a. \( 3x + 9 - \frac{20}{x - 3} \) b. \( 3x + 9 + \frac{34}{x - 3} \) c. \( 3x + 43 \)d. \( 3x^{2} + \frac{9}{x - 3} \) The correct answer is option b: \( 3x + 9 + \frac{34}{x - 3} \).

Key concepts

SAT Math Problems

The SAT math section often includes questions on a range of topics, including algebra, geometry, and data analysis. One type of problem you might encounter involves polynomial long division.
This is a process that is crucial for simplifying rational functions and solving complex algebraic expressions.
An example problem could be finding an equivalent expression for a given rational function through polynomial long division, just like the exercise we explored earlier.
In the given example, you needed to perform polynomial long division to simplify the rational function \( y = \frac{3x^2 + 7}{x - 3} \).
Understanding each of the steps deeply and knowing how to implement them under test conditions can greatly aid you in tackling such questions.
Remember to practice these types of problems regularly, as they help sharpen your skills for test day.

Rational Functions

Rational functions are a type of function expressed as the ratio of two polynomials.
An example is \( y = \frac{3x^2 + 7}{x - 3} \).
Rational functions can have asymptotes, points where the function is undefined, often occurring when the denominator equals zero.
When dealing with rational functions, it's important to simplify them whenever possible.
This can be achieved using polynomial long division or factoring, depending on the situation.
In our example, polynomial long division was used to rewrite the given rational function in a more understandable form, which ultimately helped in identifying the equivalent expression.
Algebraic manipulation and simplification are vital skills for working with rational functions efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols.
They can represent real-world quantities and relationships.
Learning to manipulate them is a key skill in algebra and is essential for solving equations and simplifying complex expressions.
In the example given, you started with the expression \( \frac{3x^2 + 7}{x - 3} \), and using polynomial long division, you simplified it to \( 3x + 9 + \frac{34}{x - 3} \).
This conversion allowed you to identify the correct equivalent expression from the given options.
Being proficient in handling algebraic expressions not only helps in academic problems but also in real-life problem-solving scenarios where variables and quantities are involved.