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Chapter 5: Problem 20

Divide each polynomial by 3 \(x^{2}\) $$ 24 x^{6}-12 x^{5}+30 x^{4} $$

Short Answer

Expert verified
8x^4 - 4x^3 + 10x^2

Step by step solution

01

- Write down the problem

Start by writing down the problem we need to solve: \( \frac{24x^{6} - 12x^{5} + 30x^{4}}{3x^{2}} \)
02

- Divide each term individually

Divide each term in the polynomial \(24x^{6} - 12x^{5} + 30x^{4}\) by \(3x^{2}\). Calculate: 1. \( \frac{24x^{6}}{3x^{2}} \) 2. \( \frac{12x^{5}}{3x^{2}} \) 3. \( \frac{30x^{4}}{3x^{2}} \)
03

- Simplify each fraction

Simplify each term from previous step: 1. \( \frac{24x^{6}}{3x^{2}} = \frac{24}{3} \times \frac{x^{6}}{x^{2}} = 8x^4 \) 2. \( \frac{12x^{5}}{3x^{2}} = \frac{12}{3} \times \frac{x^{5}}{x^{2}} = 4x^3 \) 3. \( \frac{30x^{4}}{3x^{2}} = \frac{30}{3} \times \frac{x^{4}}{x^{2}} = 10x^2 \)
04

- Combine the simplified terms

Combine the simplified terms: \(8x^4 - 4x^3 + 10x^2\)

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Key Concepts

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

simplifying polynomials
When simplifying polynomials, it鈥檚 all about breaking down the expression into more manageable parts. A polynomial is an expression made up of adding and subtracting terms, with each term being a coefficient multiplied by a variable raised to a power. Simplifying polynomials often involves combining these like terms and performing basic algebraic operations. In our problem, the polynomial is 24x鈦� - 12x鈦� + 30x鈦�. Simplifying helps to make the expression easier to work with and provides a clearer view of the polynomial's behavior.
dividing terms individually
Dividing terms individually is a strategic method to handle complex polynomial division. It means you take each term in the polynomial and divide it separately by the divisor. In our exercise with the polynomial 24x鈦� - 12x鈦� + 30x鈦� divided by 3x虏, here鈥檚 how you would approach it:

1. Divide the first term: 24x鈦� by 3x虏.
2. Divide the second term: -12x鈦� by 3x虏.
3. Divide the third term: 30x鈦� by 3x虏.

By splitting the polynomial, the task becomes much simpler, making it easy to calculate and understand each step.
combining like terms
Combining like terms is crucial when simplifying expressions. Like terms in a polynomial are the terms that have the same variable raised to the same power. For instance, in the expression 24x鈦� - 12x鈦� + 30x鈦�, each term is distinct initially. But after dividing and simplifying, the result is 8x鈦� - 4x鲁 + 10x虏.

Here鈥檚 how you combine them:
  • Simplify each term individually.
  • Add or subtract the simplified terms to get a final, more compact expression.
This process makes the polynomial more straightforward and easier to use in further calculations or expressions.

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Most popular questions from this chapter

Use scientific notation to calculate the answer to each problem. (a) The distance to Earth from Pluto is \(4.58 \times 10^{9} \mathrm{km} .\) In April \(1983,\) Pioneer 10 transmitted radio signals from Pluto to Earth at the speed of light, \(3.00 \times 10^{5} \mathrm{km}\) per sec. How long (in seconds) did it take for the signals to reach Earth? (b) How many hours did it take for the signals to reach Earth?

Evaluate. $$ 10,000(36.94) $$

Simplify by writing each expression with positive exponents. Assume that all variables represent nonzero real numbers. See Example \(5 .\) $$ \frac{\left(m^{7} n\right)^{-2}}{m^{-4} n^{3}} $$

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ \left(2 x+\frac{2}{3} y\right)\left(3 x-\frac{3}{4} y\right) $$

Find each product. \((-2 k+1)\left(8 k^{2}+9 k+3\right)\)
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