Problem 45 - Elementary and Intermediate Algebra

Question

Perform the operations. See Example 4. $$ 2\left(x^{2}+7 x-1\right)-3\left(x^{2}-2 x+2\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(-x^2 + 20x - 8\).

Step 1: Distribute the Constants

First, distribute the 2 across the first polynomial and the -3 across the second polynomial. This means you will multiply each term inside the parentheses by the number outside:\[2(x^2 + 7x - 1) = 2x^2 + 14x - 2\]\[-3(x^2 - 2x + 2) = -3x^2 + 6x - 6\]

Step 2: Combine the Like Terms

Now, combine the resulting expressions from Step 1:\[(2x^2 + 14x - 2) + (-3x^2 + 6x - 6)\]Combine the like terms:- For \(x^2\) terms: \(2x^2 - 3x^2 = -x^2\)- For \(x\) terms: \(14x + 6x = 20x\)- For constant terms: \(-2 - 6 = -8\)After combining, we get:\[-x^2 + 20x - 8\]

Step 3: Final Expression

The final result after performing the operations and combining like terms is:\[-x^2 + 20x - 8\]

Key Concepts

Distributive Property

In mathematics, the distributive property is a fundamental concept used to simplify expressions and equations. It allows us to distribute a number across terms inside parentheses. By doing this, we multiply the number outside the parentheses by each term inside. For instance, given an expression like \(2(x^2 + 7x - 1)\), we apply the distributive property by multiplying each term within the parentheses by 2:

• \(2 \cdot x^2 = 2x^2\)
• \(2 \cdot 7x = 14x\)
• \(2 \cdot (-1) = -2\)

Ultimately, applying the distributive property helps us to "break down" expressions into simpler terms, leading to a more manageable equation. By tackling each component separately, you simplify the arithmetic, making the expression easier to handle in subsequent steps.

Combining Like Terms

Combining like terms is a technique used to simplify expressions or equations by merging terms that have identical variables raised to the same power. When we say "like terms," we are referring to terms that have the same variable with the same exponent. This process helps in reducing the complexity of polynomial expressions.Consider the expression resulting from distribution: \((2x^2 + 14x - 2) + (-3x^2 + 6x - 6)\). To combine like terms, follow these steps:

• Look for terms with \(x^2\): \(2x^2 - 3x^2 = -x^2\).
• Look for terms with \(x\): \(14x + 6x = 20x\).
• Combine the constant terms: \(-2 - 6 = -8\).

By merging these like terms, the expression becomes much simpler, facilitating further operations or providing a clearer picture of the mathematical scenario delineated by the polynomial.

Simplifying Expressions

Simplifying expressions is about making a complicated expression easier and more readable while preserving its value. This involves removing unnecessary complexity and making calculations straightforward.Once you've distributed and combined like terms, what you're left with is a simplified version of the original expression. From our problem, after using the distributive property and combining like terms, the simplification of the terms gives us:\[-x^2 + 20x - 8\]Simplified expressions are the final step in solving polynomial operations, bringing together all the previous steps into a clean, easily understandable form. This process streamlines the equation, serving clarity and providing a solution that can be used in further mathematical problem-solving or analysis. Simplifying expressions not only aids in clearer insight but also ensures that further manipulations or evaluations are efficient.