Problem 23 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{2 x^{2}+3 x}{x^{2}-9 x+20}+\frac{2 x^{2}-3}{x^{2}-9 x+20}-\frac{4 x^{2}+2 x+1}{x^{2}-9 x+20}$$

Step-by-Step Solution

Answer

\(\frac{x - 4}{x^2 -9x + 20}\)

Step 1: Combine the fractions

Since all three fractions have the same denominator, they can be combined into one fraction by adding or subtracting the numerators accordingly. The resulting fraction is: \[\frac{(2x^2 + 3x) + (2x^2 -3) - (4x^2+2x+1)}{x^2 -9x +20}\]

Step 2: Simplify the numerator

Simplify the numerator by collecting like terms: \[2x^2 + 2x^2 - 4x^2 + 3x - 2x -3 -1 = 0x^2 + x -4 = x -4\]

Step 3: Final Simplification

After simplifying the numerator, the final fraction is \[\frac{x -4}{x^2 -9x +20}.\] This is the simplified form of the given expression.

Key Concepts

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form. When fractions have the same denominator, you can directly combine them. Add or subtract the numerators while keeping the common denominator. For example, with the expression \(\frac{2x^2 + 3x}{x^2 -9x +20} + \frac{2x^2 -3}{x^2 -9x +20} - \frac{4x^2 + 2x + 1}{x^2 -9x +20}\), since all fractions share the same denominator, they can be merged into one fraction.

• Combine the numerators: \((2x^2 + 3x) + (2x^2 - 3) - (4x^2 + 2x + 1)\).
• Simplified, this becomes \(\frac{x - 4}{x^2 - 9x + 20}\).

Simplification usually involves combining like terms or canceling out completely, making the fraction easier to understand and work with. If the numerator or denominator can be factored or divided by the same number, further reduction might be possible.

Polynomial Expressions

Polynomial expressions consist of variables raised to whole number powers and their coefficients. In the given exercise, the expression has terms like \(2x^2\), \(3x\), and constant numbers. Key aspects of polynomials are:

• Each term is a product of a constant and a variable raised to a power.
• Terms are usually organized in descending order based on the power of the variable, starting from the highest power to the lowest.
• Polynomial expressions can have multiple terms which could be added, subtracted, or even multiplied.

By understanding how to work with these expressions, you can effectively simplify, solve, and manipulate algebraic fractions. When dealing with polynomial expressions, always look for opportunities to simplify by combining like terms or factoring.

Combining Like Terms

Combining like terms is a crucial part of simplifying algebraic expressions. This step involves finding terms in the expression that have the same variable raised to the same power and then performing basic operations like addition or subtraction. For example, in the equation \((2x^2 + 2x^2 - 4x^2 + 3x - 2x -3 -1)\), you group and combine:

• Combine \(2x^2\), \(2x^2\), and \(-4x^2\), which sums to \(0x^2\).
• Combine \(3x\) and \(-2x\) to get \(x\).
• Combine constants \(-3\) and \(-1\) to get \(-4\).

This results in \((x - 4)\). Combining like terms reduces the complexity of expressions, making them much more straightforward to solve or analyze. Always remember to check for like terms when working through algebraic problems for a smooth and accurate simplification process.