Question

\(\left(x^{2}-4 x+3\right)-\left(3 x^{2}-4 x-3\right)\) is equivalent to: F. \(2 x^{2}-6\) G. \(2 x^{2}-8 x\) H. \(2 x^{2}-8 x-6\) J. \(-2 x^{2}+6\) K. \(-2 x^{2}-8 x\)

Short answer

A. \(4x^2 - 6\) B. \(4x^2 + 6\) C. \(-2x^2 - 6\) D. \(2x^2 + 6\) E. \(4x^2\) F. \(-2x^2 + 6\) Answer: F. \(-2x^2 + 6\)

Step-by-step solution

Remove the parentheses

To remove the parentheses, we distribute a negative sign to the second expression since it has a negative sign outside: \((x^2 - 4x + 3) - (3x^2 - 4x - 3) = x^2 - 4x + 3 - 3x^2 + 4x + 3\)

Combine like terms

Combine terms with the same variables and their degrees: \(x^2 - 3x^2 - 4x + 4x + 3 + 3 = -2x^2 + 6\)

Compare the result with the given options

By comparing the result, we can see that the equivalent expression is given by option F. \(-2x^2 + 6\) The correct answer is F. \(2x^2 - 6\).

Key concepts

Polynomial Simplification

Understanding the simplification of polynomials is essential for solving algebraic problems efficiently. Simplifying a polynomial involves reducing it to its simplest form while maintaining its equivalence. This process generally includes removing parentheses and combining like terms.

In the given exercise, the polynomial \( (x^2-4x+3) - (3x^2-4x-3) \) requires simplification to find the equivalent expression. The first step is to distribute the negative sign to every term in the second polynomial inside the parentheses. This is crucial because neglecting this step could lead to incorrect signs for the terms, altering the final outcome.

After distributing the negative sign, we get a clearer view of the polynomial without parentheses: \( x^2 - 4x + 3 - 3x^2 + 4x + 3 \) which prepares us for the next step—combining like terms. Typically, polynomial simplification questions on the ACT Math Practice test check for understanding of these essential algebraic maneuvers.

Combining Like Terms

One of the most common steps in algebra is to combine like terms. Like terms are terms that have exactly the same variable raised to the exact same power. They are the building blocks of algebraic expressions and understanding how to manage them is crucial.

To combine like terms effectively, look at each term's variable and its exponent. In our exercise, after removing the parentheses, we combine the terms with the variable \(x^2\) and then the ones with the variable \(x\), and finally, we add the constants. We get: \(x^2 - 3x^2 + (-4x + 4x) + (3 + 3)\), which simplifies to \( -2x^2 + 6 \) as constants and like terms cancel each other out.

The simplification process is not just about combining numeric coefficients; it ensures that each type of term is grouped properly to lay bare the most reduced form of the polynomial. This skill is tested in the ACT Math Practice to ensure students can manipulate algebraic expressions with precision.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent relationships and can be as simple as a single term or as complex as a long polynomial. Being proficient in handling algebraic expressions is a foundational skill in mathematics, and it is a frequent subject in ACT Math Practice questions.

When working with algebraic expressions, it's crucial to understand that each expression tells a story—how quantities relate to one another. The expression from the exercise \( -2x^2 + 6 \) indicates that the quadratic term \( x^2 \) is multiplied by -2 and then increased by a constant of 6. This simplified expression is the result of careful simplification, preserving the relationships between the quantities throughout the process.

Students must recognize that simplifying algebraic expressions is not just a mechanical process but also a way to reveal the inherent properties of the mathematical relationships they study. Mastery of algebraic expressions leads to greater success in more advanced concepts in mathematics.