Problem 10 - Algebra 2

Question

STANDARDIZED TEST PRACTICE Which expression is equal to \(\left(x^{2}-4 x+6\right)(x-3)^{-1} ?\) $$ \begin{array}{llll}{\text { A } x-1} & {\text { B } x-1+\frac{3}{x-3}} & {\text { C } x-1-\frac{3}{x-3}} & {\text { D }-x+1-\frac{3}{x-3}}\end{array} $$

Step-by-Step Solution

Answer

Option B: \(x - 1 + \frac{3}{x-3}\).

Step 1: Rewrite the Expression

Given the expression \((x^2 - 4x + 6)(x-3)^{-1}\), we need to rewrite it by multiplying \((x^2 - 4x + 6)\) by the reciprocal of \((x-3)\). This becomes \((x^2 - 4x + 6)/(x-3)\).

Step 2: Perform Polynomial Long Division

Divide \(x^2 - 4x + 6\) by \(x-3\) using polynomial long division.1. Divide the leading term \(x^2\) by \(x\) to get \(x\).2. Multiply \(x\) by \(x-3\) to get \(x^2 - 3x\).3. Subtract \(x^2 - 3x\) from \(x^2 - 4x + 6\) to get \(-x + 6\).4. Divide \(-x\) by \(x\) to get \(-1\).5. Multiply \(-1\) by \(x-3\) to get \(-x + 3\).6. Subtract \(-x + 3\) from \(-x + 6\) to get the remainder of \(3\).The quotient is \(x - 1\) with a remainder of \(3\), written as \(x-1 + \frac{3}{x-3}\).

Step 3: Compare with Answer Choices

The expression obtained from the division \(x - 1 + \frac{3}{x-3}\) matches with option B in the given answer choices.

Key Concepts

Rational Expressions

Rational expressions are like fractions, but instead of just numbers in the numerator and denominator, they have polynomials. It's important to understand how to manipulate these expressions because they frequently appear in algebra.
When working with rational expressions, you often need to simplify them by factoring and canceling common factors from the numerator and the denominator.
Another critical aspect is understanding how to perform operations such as addition, subtraction, multiplication, and division on these expressions.

• For addition and subtraction, find a common denominator.
• For multiplication, multiply numerators together and denominators together.
• For division, multiply by the reciprocal of the divisor - just as seen in the problem we solved.

Polynomial long division is often used when dealing with more complex rational expressions, such as dividing one polynomial by another. This process helps to break down the division step into something manageable, leading us to an expression that easily matches an answer choice.

Arithmetic with Polynomials

Arithmetic with polynomials involves performing operations such as addition, subtraction, multiplication, and division, much like the arithmetic we do with numbers. Understanding these operations is essential for solving problems with polynomials.
One of the most common operations is polynomial long division, a method similar to long division with numbers. It helps break down a division problem between two polynomials into simpler parts.
Here's a quick rundown of polynomial division:

• Align terms in a standard form, from highest to lowest degree.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract from the original dividend.
• Repeat the process with the new polynomial formed.

In our exercise, using polynomial long division was crucial to simplifying the rational expression and matching it to the correct answer choice.

Standardized Test Practice

Standardized tests often include questions on rational expressions and polynomial arithmetic to assess your understanding and mastery of these concepts. Success in these tests requires familiarity with various algebraic techniques.
During test practice, pay attention to:

• Time management: Don't spend too much time on one problem.
• Recognizing patterns: Familiarity with common problem types can speed up the solving process.
• Checking work: Review steps to avoid careless mistakes, especially in intricate polynomial divisions.

In the context of our exercise, recognizing that the division form presented an opportunity to apply polynomial long division adequately was key to finding the correct answer choice quickly. This skill becomes particularly useful in standardized tests where efficiency is as important as accuracy.