Problem 95 - Intermediate Algebra

Question

Simplify each expression. \(2\left(x^{2}+4 x-1\right)+3\left(2 x^{2}-2 x+2\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(8x^2 + 2x + 4\).

Step 1: Expand the first expression

Multiply each term inside the parenthesis \((x^2 + 4x - 1)\) by 2: \[2(x^2 + 4x - 1) = 2x^2 + 8x - 2\]

Step 2: Expand the second expression

Multiply each term inside the parenthesis \((2x^2 - 2x + 2)\) by 3: \[3(2x^2 - 2x + 2) = 6x^2 - 6x + 6\]

Step 3: Combine like terms

Add the expanded expressions together and combine like terms: \[(2x^2 + 8x - 2) + (6x^2 - 6x + 6) = (2x^2 + 6x^2) + (8x - 6x) + (-2 + 6)\] Simplify each set of like terms:\[8x^2 + 2x + 4\]

Key Concepts

Polynomial Expansion

Polynomial expansion is a process where you remove the parentheses in an expression by distributing or multiplying each term inside the parentheses by the factor outside. This process helps in breaking down complex expressions into simpler ones, making them easier to solve or simplify later.
To perform polynomial expansion, you need to follow these steps:

• Identify the coefficient or number outside the parentheses.
• Multiply this coefficient to each term inside the parentheses individually.
• Write down the new expression created after multiplication.

In the given exercise, we applied this to both terms in the expression. We distributed 2 to the terms inside the first set of parentheses and 3 to the terms inside the second set. This gives us: \[2(x^2 + 4x - 1) = 2x^2 + 8x - 2\] and \[3(2x^2 - 2x + 2) = 6x^2 - 6x + 6\].
This step is fundamental as it sets the stage for combining like terms.

Combining Like Terms

Once polynomial expansion is complete, the next step is to simplify the expression further by combining like terms. Like terms are terms that have the same variable raised to the same power. While the coefficients can be different, the variable and its power must match.
Here's how to combine like terms:

• Identify terms in the expression with the same variable and power.
• Add or subtract their coefficients.
• Write the simplified expression with the combined terms.

In our expression, after expansion, we have like terms such as \[2x^2\] and \[6x^2\], making \[8x^2\]. Likewise, \[8x\] and \[-6x\] combine to form \[2x\]. Finally, the constants \[-2\] and \[+6\] sum up to \[4\].
This approach simplifies the expression and reduces it to its simplest form.

Algebraic Simplification

Algebraic simplification is about making an expression as concise and manageable as possible while ensuring it maintains the same value. The objective is to reduce the expression to a form where no further simplification is possible, presenting it in its simplest form.
Simplification often involves:

• Expanding expressions via distribution (as seen in polynomial expansion).
• Combining like terms to reduce complexity.
• Eliminating unnecessary terms or factors if possible.

In this exercise, algebraic simplification allowed us to transform a lengthy expression involving multiple terms and parentheses into \[8x^2 + 2x + 4\]. It’s crucial because it facilitates easier calculations and better understanding in solving algebra problems.
Simplified expressions are easier to evaluate or substitute with values, providing clearer insights into the mathematical relationships they describe.