Q: EXAMPLES 11 + x ______ x^3 + 2x(5 - x)

Answer: The simplified form of the expression is $\frac{11 + x}{x^3} + 10 - 2x$. Explanation: This problem involves algebraic simplification, specifically combining rational expressions and polynomial terms. The key concepts include the properties of fractions, distributive property, and combining like terms. The goal is to write the expression in a simplified, more manageable form. Steps: Original expression: \[ \frac{11 + x}{x^3} + 2x(5 - x) \] Distribute $2x$ in the second term: \[ 2x \times 5 = 10x \] \[ 2x \times (-x) = -2x^2 \] So, the expression becomes: \[ \frac{11 + x}{x^3} + 10x - 2x^2 \] Rewrite the entire expression clearly: \[ \frac{11 + x}{x^3} + 10x - 2x^2 \] Optional: Express all terms with common denominators if further simplification is desired. Since the first term has denominator $x^3$, rewrite the polynomial terms as fractions with denominator $x^3$: \[ 10x = \frac{10x \times x^3}{x^3} = \frac{10x^4}{x^3} \] \[ -2x^2 = \frac{-2x^2 \times x^3}{x^3} = \frac{-2x^5}{x^3} \] The entire expression becomes: \[ \frac{11 + x}{x^3} + \frac{10x^4}{x^3} - \frac{2x^5}{x^3} \] Combine into a single fraction: \[ \frac{11 + x + 10x^4 - 2x^5}{x^3} \] Final simplified form: \[ \boxed{\frac{11 + x + 10x^4 - 2x^5}{x^3}} \] Note: Depending on the context, you might leave the expression as the sum of a rational expression and polynomial terms or combine into a single fraction. The initial step of distributing and simplifying is often sufficient unless further algebraic manipulation is required.

Q: What is the number who chose Italian dressing?

Answer: The number of customers who chose Italian dressing is 90. Explanation: This problem involves understanding the distribution of salad dressing choices among customers, which is represented by a circle graph (pie chart). The key concepts involved are percentage calculations and proportions. The problem asks to find the difference in the number of customers choosing Italian dressing and vinaigrette dressing, given the total number of customers and the percentage distribution. The method involves: Calculating the number of customers for each dressing based on the percentage. Finding the difference between the two numbers. Steps: Identify the total number of customers: Total customers = 200 Determine the percentage for Italian dressing: From the table, Italian dressing = 45% Calculate the number of customers who chose Italian dressing: \[ \text{Number of Italian dressing customers} = \frac{45}{100} \times 200 = 0.45 \times 200 = 90 \] Determine the percentage for Vinaigrette dressing: Vinaigrette = 30% Calculate the number of customers who chose Vinaigrette dressing: \[ \frac{30}{100} \times 200 = 0.30 \times 200 = 60 \] Find the difference between Italian and Vinaigrette dressing customers: \[ 90 - 60 = 30 \] Question asks for the difference between the number who chose Italian dressing and the number who chose vinaigrette dressing: The difference is 30. Note: The question also mentions the number who chose vinaigrette dressing, but the main calculation is to find the number for Italian dressing, which is 90.

Question

EXAMPLES 11 + x ______ x^3 + 2x(5 - x)

Accepted Answer

Answer: The simplified form of the expression is $\frac{11 + x}{x^3} + 10 - 2x$.

Explanation:  This problem involves algebraic simplification, specifically combining rational expressions and polynomial terms. The key concepts include the properties of fractions, distributive property, and combining like terms. The goal is to write the expression in a simplified, more manageable form.

Steps:

Original expression:

$$
   \frac{11 + x}{x^3} + 2x(5 - x)
   $$

Distribute $2x$ in the second term:

$$
   2x \times 5 = 10x
   $$   $$
   2x \times (-x) = -2x^2
   $$   So, the expression becomes:     $$
   \frac{11 + x}{x^3} + 10x - 2x^2
   $$

Rewrite the entire expression clearly:

$$
   \frac{11 + x}{x^3} + 10x - 2x^2
   $$

Optional: Express all terms with common denominators if further simplification is desired.

Since the first term has denominator $x^3$, rewrite the polynomial terms as fractions with denominator $x^3$:     $$
   10x = \frac{10x \times x^3}{x^3} = \frac{10x^4}{x^3}
   $$   $$
   -2x^2 = \frac{-2x^2 \times x^3}{x^3} = \frac{-2x^5}{x^3}
   $$

The entire expression becomes:     $$
   \frac{11 + x}{x^3} + \frac{10x^4}{x^3} - \frac{2x^5}{x^3}
   $$

Combine into a single fraction:

$$
   \frac{11 + x + 10x^4 - 2x^5}{x^3}
   $$

Final simplified form:

$$
   \boxed{\frac{11 + x + 10x^4 - 2x^5}{x^3}}
   $$

Note: Depending on the context, you might leave the expression as the sum of a rational expression and polynomial terms or combine into a single fraction. The initial step of distributing and simplifying is often sufficient unless further algebraic manipulation is required.

EXAMPLES 11 + x ______ x^3 + 2x(5 – x)