Lesson 4: Distribution and Factoring — Practice Questions

1. 1. Simplify the fraction by factoring and canceling common factors: $$\frac{5x + 15}{7x + 21}$$ The simplified form is ___.

2. 2. Reduce the following fraction to its simplest form: $$\frac{10x^3}{15x}$$ The simplified form is ___.

3. 3. Which of the following is the simplified form of the expression $$\frac{x+5}{x+10}$$?

   • A. $\frac{1}{2}$
   • B. $\frac{x+1}{x+2}$
   • C. $\frac{1}{x+2}$
   • D. $\frac{x+5}{x+10}$

4. 4. Simplify the following rational expression completely: $$\frac{14x - 28}{21x - 42}$$ The result is ___.

5. 5. When simplifying the fraction $\frac{8x + 16}{4x}$, what is the correct first step?

   • A. Cancel the $x$ from $8x$ and $4x$.
   • B. Divide 16 by 4.
   • C. Factor the numerator into $8(x+2)$.
   • D. Subtract $4x$ from $8x$.

6. 6. Which of the following expressions can be simplified by canceling the number 3?

   • A. $\frac{x+3}{y+3}$
   • B. $\frac{3x+1}{3}$
   • C. $\frac{3(x-2)}{3}$
   • D. $\frac{x-3}{3}$

7. 7. Simplify the expression $\frac{6a}{6b+12}$ by first factoring the denominator and then canceling the common factor. The simplified form is ___.

8. 8. Is the statement $\frac{y+10}{z+10} = \frac{y}{z}$ a correct simplification?

   • A. Yes, because the 10s can be canceled.
   • B. No, because 10 is a term, not a factor.

9. 9. Simplify the rational expression $\frac{4(x-7)}{8x-56}$. The simplified form is ___.

10. 10. Why can the expression $\frac{7y+4}{7}$ not be simplified to $y+4$?

    • A. Because 7 is not a factor of the entire numerator.
    • B. Because variables cannot be part of a cancellation.
    • C. Because 4 is not divisible by 7.
    • D. Because addition must be performed before division.