Problem 19 - Intermediate Algebra

Question

Perform each division. \(\frac{4 x^{4}+6 x}{2}\)

Step-by-Step Solution

Answer

The result is \(2x^4 + 3x\).

Step 1: Rewrite the Division Expression

Rewrite the expression to distribute the denominator to each term in the numerator. The given expression is \(\frac{4x^4 + 6x}{2}\), which can be rewritten as \(\frac{4x^4}{2} + \frac{6x}{2}\).

Step 2: Simplify the First Term

Divide the first term of the numerator by the denominator. Simplify \(\frac{4x^4}{2}\) by dividing the coefficients and subtracting exponents: \(\frac{4}{2}x^4 = 2x^4\).

Step 3: Simplify the Second Term

Divide the second term of the numerator by the denominator. Simplify \(\frac{6x}{2}\) by dividing the coefficients: \(\frac{6}{2}x = 3x\).

Step 4: Combine the Simplified Terms

Combine the simplified terms to write the final expression. After simplifying each term separately, the final expression is \(2x^4 + 3x\).

Key Concepts

Algebraic Expressions

Algebraic expressions form the backbone of algebra and are used to express mathematical ideas and relationships. An algebraic expression can include numbers, variables, and operations. Consider it a mathematical phrase composed of:

• Variables (like \(x, y\)) which are symbols representing unknown values.
• Constants, which are numbers without variables attached (like 2 or 3).
• Operations, including addition, subtraction, multiplication, and division.

Algebraic expressions can be simple, like \(3x + 5\), or more complex, like \(4x^4 + 6x - 7\). These expressions allow us to formulate equations, represent real-world problems, and perform calculations. Breaking an expression into recognizable parts makes it easier to work with, as seen in polynomial division.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more manageable or simpler form without changing their value. In the context of polynomial division, like in \(\frac{4x^4 + 6x}{2}\), simplification helps reduce the expression to the smallest terms.

• Distribute the division across individual terms: Rewrite the entire expression by applying the divisor to each term in the numerator.
• Reduce the coefficients: Divide the numbers in front of the variables (coefficients) by the divisor, as well as apply any simplification rules for variables.

This process results in a simpler, more straightforward expression that's easier to work with. For example, \(\frac{4x^4}{2}\) simplifies to \(2x^4\) and \(\frac{6x}{2}\) simplifies to \(3x\), making the overall expression \(2x^4 + 3x\).

Exponents

Exponents are a powerful tool in mathematics, representing repeated multiplication of a number by itself. They appear frequently in algebraic expressions, such as \(x^4\), meaning \(x\) multiplied by itself four times.

• An exponent tells us how many times to use the base in a multiplication. For example, \(x^3 = x \cdot x \cdot x\).
• When simplifying expressions, you can use exponent rules, such as multiplying or dividing powers with the same base by adding or subtracting the exponents, respectively.

For instance, in the step where \(\frac{4x^4}{2}\) is simplified, the exponent remains unchanged as we're dividing by a constant (not another power of \(x\)). These properties are crucial in simplifying expressions and solving equations efficiently.