Math  /  Algebra

QuestionConsider the expression 2x3+5x2+5x3x3+1\frac{2 x^{3}+5 x^{2}+5 x}{3 x^{3}+1}.
Part: 0/30 / 3
Part 1 of 3 (a) Divide the numerator and denominator by the greatest power of xx that appears in the denominator. That is, divide each term in the numerator and denominator by x3x^{3}. Write your answers with positive exponents only. 2x3+5x2+5x3x3+1=+++\frac{2 x^{3}+5 x^{2}+5 x}{3 x^{3}+1}=\frac{\square+\square+\square}{\square+\square}

Studdy Solution

STEP 1

What is this asking? We're simplifying a fraction with xxs by dividing the top and bottom by the highest power of xx in the bottom, which is x3x^3. Watch out! Don't forget to apply the division to *every single term* in both the numerator and the denominator!
Also, remember your exponent rules!

STEP 2

1. Divide the numerator terms by x3x^3.
2. Divide the denominator terms by x3x^3.

STEP 3

We have 2x32x^3.
We're dividing it by x3x^3.
So we have 2x3x3=2x3x3=21=2\frac{2x^3}{x^3} = 2 \cdot \frac{x^3}{x^3} = 2 \cdot 1 = \textbf{2}.
We divided x3x^3 by itself, which gives us 1\textbf{1}!

STEP 4

Now we have 5x25x^2.
Dividing by x3x^3 gives us 5x2x3=5x2x3=51x=5x\frac{5x^2}{x^3} = 5 \cdot \frac{x^2}{x^3} = 5 \cdot \frac{1}{x} = \frac{\textbf{5}}{\textbf{x}}.
Remember your exponent rules!

STEP 5

Lastly in the numerator, we have 5x5x.
Dividing by x3x^3 gives us 5xx3=5xx3=51x2=5x2\frac{5x}{x^3} = 5 \cdot \frac{x}{x^3} = 5 \cdot \frac{1}{x^2} = \frac{\textbf{5}}{\textbf{x}^\textbf{2}}.
Exponent rules are key!

STEP 6

We have 3x33x^3.
Dividing it by x3x^3 gives us 3x3x3=3x3x3=31=3\frac{3x^3}{x^3} = 3 \cdot \frac{x^3}{x^3} = 3 \cdot 1 = \textbf{3}.
Just like we did for the first term in the numerator!

STEP 7

Finally, we have 11.
Dividing it by x3x^3 gives us 1x3=1x3\frac{1}{x^3} = \frac{\textbf{1}}{\textbf{x}^\textbf{3}}.
Nice!

STEP 8

Putting it all together, we get: 2x3+5x2+5x3x3+1=2+5x+5x23+1x3 \frac{2x^3 + 5x^2 + 5x}{3x^3 + 1} = \frac{2 + \frac{5}{x} + \frac{5}{x^2}}{3 + \frac{1}{x^3}} So, the boxes are filled with 22, 5x\frac{5}{x}, 5x2\frac{5}{x^2}, 33, and 1x3\frac{1}{x^3}.

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