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Chapter 6: Problem 449

Divide each polynomial by the monomial. $$\left(8 x^{3}+6 x^{2}\right) \div 2 x$$

Short Answer

Expert verified
The result is \(4x^{2} + 3x\).

Step by step solution

01

Understand the Question

The objective is to divide the polynomial \(8x^{3} + 6x^{2}\) by the monomial \(2x\).
02

Distribute the Division

Divide each term of the polynomial by the monomial separately: \[ \frac{8x^{3}}{2x} + \frac{6x^{2}}{2x} \]
03

Simplify Each Term

Simplify the fractions: \[ \frac{8x^{3}}{2x} = 4x^{2} \] \[ \frac{6x^{2}}{2x} = 3x \]
04

Combine the Results

Combine the simplified terms: \[ 4x^{2} + 3x \]
05

Verify the Solution

Ensure the polynomial has been properly divided. The result should be \[ 4x^{2} + 3x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

monomial division
When dividing polynomials by a monomial, we break down the process into manageable steps. Monomial division is simply the division of a polynomial by a single term known as a monomial. Whenever we encounter such problems, we can treat each term of the polynomial separately. For instance, in the example $$ (8x^3 + 6x^2) 梅 2x $$, we will consider each term, which are $$ 8x^3 $$ and $$ 6x^2 $$, individually and divide each by the monomial $$ 2x $$.
distribute division
The next step involves distributing the division across the polynomial鈥檚 terms. This means we divide each term of $$ 8x^3 + 6x^2 $$ by the monomial $$ 2x $$ independently. Mathematically, it looks like this: $$ \frac{8x^3}{2x} + \frac{6x^2}{2x} $$. Distributing the division helps to simplify the problem. We turn a complex problem into simpler, smaller divisions that are easier to handle.
simplify fractions
Once we have distributed the division, our next task is to simplify the fractions. In simplifying each term: $$ \frac{8x^3}{2x} $$ and $$ \frac{6x^2}{2x} $$, we perform the actual division. For example: $$ \frac{8x^3}{2x} $$ simplifies to $$ 4x^2 $$ because $$ 8 梅 2 = 4 $$ and $$ x^3 梅 x = x^2 $$. Similarly, $$ \frac{6x^2}{2x} $$ breaks down to $$ 3x $$ where $$ 6 梅 2 = 3 $$ and $$ x^2 梅 x = x $$. By simplifying, we make each term of the division more straightforward.
polynomial terms
Finally, let鈥檚 combine the simplified terms. Polynomial terms are the individual components of a polynomial separated by plus or minus signs. After distributing the division and simplifying the fractions, we have: $$ 4x^2 $$ and $$ 3x $$. Combining these, we get the final result: $$ 4x^2 + 3x $$. Polynomial terms can be easily added together like this once they are simplified. Ensuring each term is correct is essential for verifying the accuracy of our division. The resulting polynomial, $$ 4x^2 + 3x $$, is the solution to our original division problem.

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Most popular questions from this chapter

Simplify. (a) \(5^{-3}\) (b) \(10^{-5}\)

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