Problem 69 - Algebra for College Students

Question

Simplify by removing the inner parentheses first and working outward. $$\left[4 x^{3}-\left(2 x^{2}-x-1\right)\right]-\left[5 x^{3}-\left(x^{2}+2 x-1\right)\right]$$

Step-by-Step Solution

Answer

\(-x^3 - x^2 + 3x\)

Step 1: Remove Inner Parentheses

Begin by dealing with the inner parentheses in both bracketed expressions. Apply the distributive property by changing the sign of each term inside the parentheses.1. In the first expression:\[ - (2x^2 - x - 1) = -2x^2 + x + 1 \]2. In the second expression:\[ - (x^2 + 2x - 1) = -x^2 - 2x + 1 \]

Step 2: Simplify Each Bracketed Expression

Next, substitute the simplified inner expressions back into their respective bracketed expressions and then simplify.1. First bracketed expression:\[ 4x^3 - 2x^2 + x + 1 \]2. Second bracketed expression:\[ 5x^3 - x^2 - 2x + 1 \]

Step 3: Remove Outer Brackets

Subtract the second simplified expression from the first, carefully considering the negative sign in front of the second expression.\[\left(4x^3 - 2x^2 + x + 1 \right) - \left(5x^3 - x^2 - 2x + 1\right)\]Change the signs of all terms inside the second expression:\[4x^3 - 2x^2 + x + 1 - 5x^3 + x^2 + 2x - 1\]

Step 4: Combine Like Terms

Combine the like terms to simplify the expression further:\[4x^3 - 5x^3 - 2x^2 + x^2 + x + 2x + 1 - 1\]Collect like terms:\[-x^3 - x^2 + 3x\]

Step 5: Final Simplified Expression

The expression is now simplified completely to the following:\[-x^3 - x^2 + 3x\]

Key Concepts

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients.They can contain operations like addition, subtraction, multiplication, and non-negative integer exponents of variables.Understanding polynomials is crucial as they are foundational in algebra and appear in various forms and equations.
Polynomials are written in terms of their degree, which is the highest exponent of the variable.For example, in the expression \[4x^3 - 2x^2 + x\],the degree of the polynomial is 3 because the highest exponent of \(x\) is 3.
Key characteristics of polynomials include:

• Terms: Building blocks of polynomials, separated by addition or subtraction.
• Coefficients: Numerical factors of each term.
• Variable: A letter representing an unknown quantity, commonly \(x\).

Understanding these features aids in mastering the simplification and operations involving polynomials.

Distributive Property

The distributive property is a fundamental algebraic principle that helps in simplifying expressions by distributing operations over terms inside parentheses.In essence, it involves multiplying a single term outside the parentheses by each term inside and changing signs if subtraction is applied.
In the step-by-step solution of our exercise, the distributive property was employed to simplify expressions within the inner parentheses:

• For \(-(2x^2 - x - 1)\), each term inside is affected by the negative sign: \[-2x^2 + x + 1\]
• Similarly, for \(-(x^2 + 2x - 1)\), distributing changes the terms to: \[-x^2 - 2x + 1\]

This technique ensures that all operations are applied correctly to each term, making sure the expression is ready for further simplification.
Mastery of the distributive property is essential for tackling complex algebraic expressions, ensuring clarity and correctness in problem-solving.

Combining Like Terms

Combining like terms is a simple yet crucial algebraic process used to simplify equations.It involves adding or subtracting coefficients of terms that have the exact same variables raised to the same power.This step is usually performed after removing parentheses and applying the distributive property.
In the given exercise, after using the distributive property, we identify and combine like terms:

• The terms \(4x^3\) and \(-5x^3\) are combined to give \(-x^3\).
• Similarly, \(-2x^2\) and \(x^2\) are combined as \(-x^2\).
• The terms \(x\) and \(2x\) result in \(3x\).

By combining like terms, you reduce the complexity of the expression, moving one step closer to the solution.
This method is instrumental in solving algebraic equations, as it helps to streamline the expression, making it easier to analyze and solve.