Problem 17 - Elementary Algebra

Question

\(\left(5 w^{4}-15 w^{2}+60 w+20\right) \div(5 w)\)

Step-by-Step Solution

Answer

w^3 - 3w + 12 + \frac{4}{w}

Step 1: Distribute the Division

Divide each term in the polynomial \(5w^4 - 15w^2 + 60w + 20\) by \(5w\). This means we separately divide \(5w^4\) by \(5w\), \(-15w^2\) by \(5w\), \(60w\) by \(5w\) and \(20\) by \(5w\).

Step 2: Divide Each Term

Calculate each individual division: \(\frac{5w^4}{5w} = w^3 \), \(\frac{-15w^2}{5w} = -3w \), \(\frac{60w}{5w} = 12 \) and \(\frac{20}{5w} = \frac{4}{w}\).

Step 3: Combine Results

Combine the results from step 2 to write the simplified form of the expression: \(w^3 - 3w + 12 + \frac{4}{w}\).

Key Concepts

Polynomial Simplification

To simplify a polynomial, we break it down into simpler parts. This is important in algebra because it makes equations easier to solve and understand. In our exercise, we started with the polynomial \(5w^4 - 15w^2 + 60w + 20\).

By distributing the division across all terms in the polynomial, we simplified each term separately. Reducing complex polynomials helps in many areas of math, like solving quadratic equations or integrating functions. Remember: always ensure that each term is divided and simplified properly to avoid mistakes.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. In our exercise, we had to simplify an algebraic expression by dividing. This process involves understanding the role of each component in the expression.

For example:

• Numbers: like 5, 15, 60, 20 in our polynomial.
• Variables: the letter w represents an unknown value.
• Operations: in this case, we used division.

Combining these elements in a correct and simplified form such as \(w^3 - 3w + 12 + \frac{4}{w}\), makes the expression manageable, which is crucial in real-world problem-solving.

Division of Terms

Dividing terms in a polynomial involves breaking down each term individually. This means performing the division operation on each term within the polynomial separately.

In our exercise, terms like \(5w^4\) and \(-15w^2\) were divided by \(5w\) one by one. For instance, \(\frac{5w^{4}}{5w} = w^{3}\). This step-by-step approach ensures accuracy in simplifying the entire expression.

The division of terms doesn't just stop at coefficients, it also applies to variables, which makes it essential to understand the properties of exponents and how they interact during division. This helps to break down even the most complex polynomials.