Problem 41 - Algebra 1

Question

Simplify the expression. $$8 x^{2} \div\left(-\frac{4}{5}\right)$$

Step-by-Step Solution

Answer

-10 x^{2}

Step 1: Understand the Problem

We're asked to simplify the expression \(8 x^{2} \div (-\frac{4}{5})\). To do this, remember that dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Rewrite the Division as Multiplication

We rewrite the expression as \(8 x^{2}*(-\frac{5}{4})\). This is done by dividing by \(-\frac{4}{5}\), which is equivalent to multiplying by its reciprocal, \(-\frac{5}{4}\).

Step 3: Simplify the Expression

Now, we multiply \(8 x^{2}\) by \(-\frac{5}{4}\). This gives us \(-10 x^{2}\).

Key Concepts

Division of Polynomials

Dividing polynomials might sound a bit intimidating, but it's quite straightforward once you get the hang of it. To divide a polynomial by a number or another polynomial, you just need to apply some basic arithmetic principles.
The main idea here is similar to dividing regular numbers. For example, when you see a polynomial expression like \(8x^{2} \div -\frac{4}{5}\), you can interpret this as asking how many times \(-\frac{4}{5}\) can be taken out from \(8x^{2}\).

However, in algebra, especially with complex polynomials, it’s common to see fractions involved. This is where dividing by a fraction becomes more like a puzzle. The key rule to remember is that dividing by a fraction is the same as multiplying by its reciprocal. This makes the division process a lot easier as you transform the problem into a multiplication problem.

Fractions in Algebra

Fractions often play a critical role in algebra, especially when dealing with expressions and equations. Understanding how fractions work within algebraic expressions is crucial.
When you see a fraction in algebra like \(-\frac{4}{5}\), remember that it represents division: the numerator \(-4\) divided by the denominator \(5\).

• To make computations involving fractions more manageable, converting division into multiplication by using the reciprocal can simplify the problem.
• For example, instead of dividing by \(-\frac{4}{5}\), you multiply by its reciprocal \(-\frac{5}{4}\).

This technique helps maintain clarity in your calculations and avoids confusion when simplifying complex expressions. Such transformations are a powerful tool to streamline and simplify algebraic work.

Simplification Techniques

Simplifying algebraic expressions is a fundamental skill that enhances your ability to solve equations efficiently. The goal of simplification is to express the algebraic equation or term in the simplest form possible.
Here's a simple step to effectively simplify the given polynomial division:

• After rewriting the division as multiplication (i.e., \(8x^{2} * -\frac{5}{4}\)), you can directly multiply the terms.
• First, multiply the coefficients: \(8\) and \(-\frac{5}{4}\), giving \(-10\).
• Next, multiply the variable part: \(x^{2}\). As there are no other variable terms in this multiplication, you simply retain \(x^{2}\).

Merging these results, you get the simplified expression: \(-10x^{2}\). Such simplification not only provides the final answer but also reduces complexity, making the solution clear and manageable.