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Chapter 5: Problem 7

Perform each division. $$ \frac{60 x^{4}-20 x^{2}+10 x}{2 x} $$

Short Answer

Expert verified
30x^3 - 10x + 5

Step by step solution

01

Divide each term by the denominator

The given expression is \[ \frac{60 x^{4}-20 x^{2}+10 x}{2 x} \]. Divide each term in the numerator by the denominator separately: \[ \frac{60 x^{4}}{2 x} - \frac{20 x^{2}}{2 x} + \frac{10 x}{2 x} \]
02

Simplify each fraction

- For \[ \frac{60 x^{4}}{2 x} \]: Divide the coefficients (60 梅 2) and subtract the exponents of x (4 - 1), resulting in \[ 30 x^{3} \]. - For \[ \frac{20 x^{2}}{2 x} \]: Divide the coefficients (20 梅 2) and subtract the exponents of x (2 - 1), resulting in \[ 10 x \]. - For \[ \frac{10 x}{2 x} \]: The x terms cancel each other out, giving \[ 5 \].
03

Combine the simplified terms

Combine the results from Step 2: \[ 30 x^{3} - 10 x + 5 \].

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Key Concepts

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

dividing polynomials
Dividing polynomials might seem tricky, but you just need to follow a few simple steps. Start with a polynomial (the numerator) you need to divide by another polynomial (the denominator). The key is breaking it down term by term to make things manageable.
The first step is to divide each term in the numerator by the denominator separately. For instance, in our example, the polynomial \( 60 x^{4}-20 x^{2}+10 x \) was divided by the simple polynomial \( 2x \). Subdivide the terms of the numerator:
  • \( \frac{60 x^{4}}{2 x} \)
  • \( \frac{20 x^{2}}{2 x} \)
  • \( \frac{10 x}{2 x} \)

By treating each term individually, the problem becomes easier to handle.
simplifying expressions
Once you have divided each term, the next task is simplification. This step involves breaking down each fraction by handling the coefficients and the variables. When you simplify, you need to:
  • Divide the numerical coefficients.
  • Adjust the exponents based on the rules for dividing like terms.

For example:
  • \( \frac{60 x^{4}}{2 x} \): Divide 60 by 2 to get 30, and reduce the exponent of \( x^{4} \) by \( 1 \) to get \( x^{3} \). So, the result is \( 30 x^{3} \).
  • \( \frac{20 x^{2}}{2 x} \): Divide 20 by 2, getting 10, and reduce the exponent of \( x^{2} \) by \( 1 \) to get \( x \). This gives \( 10x \).
  • For \( \frac{10 x}{2 x} \), since the xs cancel each other, you simply get 5.

After simplifying, you combine these results: \( 30 x^{3} - 10 x + 5 \).
algebraic fractions
Algebraic fractions appear often in polynomial division exercises. They look similar to numerical fractions but include variables within them. When working with these fractions, it's essential to:
  • Simplify the coefficients just like regular numbers.
  • Apply the rules of exponents to the variables.

Remember, an algebraic fraction is simplified by reducing it as much as possible: turning complex expressions into simpler forms.
Here's how it works in practice with our example:
  • \( \frac{60 x^{4}}{2 x} \) becomes \( 30 x^{3} \)
  • \( \frac{20 x^{2}}{2 x} \) reduces to \( 10 x \)
  • \( \frac{10x}{2x} \) simplifies to \( 5 \)

Breaking down complex algebraic fractions into smaller, simpler pieces can make solving polynomial division much easier.

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Most popular questions from this chapter

Perform each division. $$ \frac{3 t^{4}+5 t^{3}-8 t^{2}-13 t+2}{t^{2}-5} $$

Evaluate. $$ 49 \div 10,000 $$

Perform each division. $$ \left(2 x^{3}+x+2\right) \div(x+1) $$

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(4 a^{2} b^{3}\right)^{-2}\left(2 a b^{-1}\right)^{3}}{\left(a^{3} b\right)^{-4}} $$

Perform each division using the "long division" process. $$ \frac{5 z^{3}-z^{2}+10 z+2}{z+2} $$
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