Rational Expression Operations - Algebra 3

1. Which expression is equivalent to $\frac{x}{x-3}-\frac{2}{x+3}$? Assume the denominators are nonzero.
   Answer: $\frac{x^2+x+6}{x^2-9}$ Whenever you see rational expressions being added or subtracted, your first move is to rewrite them with a common denominator. Here, the denominators are $x-3$ and $x+3$, so the common denominator is $(x-3)(x+3)=x^2-9$. To combine, multiply the first numerator by the factor its denominator lacks: $x(x+3)=x^2+3x$. Multiply the second numerator by the factor its denominator lacks: $2(x-3)=2x-6$. Keep the subtraction sign in front: $$\frac{x(x+3)-2(x-3)}{x^2-9} = \frac{x^2+3x-2x+6}{x^2-9} = \frac{x^2+x+6}{x^2-9}.$$ That is the expression you want. The other choices come from common errors. $\frac{x^2+5x+6}{x^2-9}$ looks like adding $2(x+3)$ instead of subtracting $2(x-3)$, so the sign and multiplier are both wrong. $\frac{x^2+x-6}{x^2-9}$ happens if you multiply the second term by $x+3$ instead of $x-3$, leaving the second denominator unchanged incorrectly. $\frac{x^2-5x+6}{x^2-9}$ comes from multiplying both terms by $x-3$, so the second term never gets the $x+3$ factor it needs. A strong takeaway: when combining rational expressions, multiply each numerator only by the factor missing from its own denominator, and distribute the negative sign carefully. One sign slip can turn $6$ into $-6$.
2. What is the simplified form of $\frac{1}{x^2+3x+2}+\frac{1}{x^2+5x+6}$?
   Answer: $\frac{2}{(x+1)(x+3)}$ Whenever you see sums of rational expressions, your first move should be to factor every denominator. Here, $x^2+3x+2=(x+1)(x+2)$ and $x^2+5x+6=(x+2)(x+3)$, so the common denominator is $(x+1)(x+2)(x+3)$. Rewriting each fraction with this denominator gives $\frac{x+3}{(x+1)(x+2)(x+3)} + \frac{x+1}{(x+1)(x+2)(x+3)}$. Adding the numerators yields $\frac{2x+4}{(x+1)(x+2)(x+3)}$. Now simplify: factor the numerator as $2(x+2)$, then cancel the common $(x+2)$ factor. This leaves $\frac{2}{(x+1)(x+3)}$, the correct simplified form. The choice $\frac{2x+4}{(x+1)(x+2)(x+3)}$ is a trap because it shows the correct sum before canceling — it is not yet simplified. The choice $\frac{2x+2}{(x+1)(x+2)(x+3)}$ likely comes from adding $x+3$ and $x+1$ incorrectly as $2x+2$ instead of $2x+4$. The choice $\frac{2}{(x+2)(x+3)}$ suggests canceling the $(x+1)$ factor entirely, which is not valid because $(x+1)$ does not appear in the numerator after simplification. Always factor first, combine over the common denominator, and then cancel common factors from numerator and denominator. A quick final check: plug in a value like $x=0$ to confirm your simplified answer matches the original expression.
3. Which expression has the same simplified form and the same domain restrictions as $\frac{x^2-1}{x^2-2x+1}$?
   Answer: $\frac{x+1}{x-1}$, with $x\neq1$ The original factors as $\frac{(x-1)(x+1)}{(x-1)^2}=\frac{x+1}{x-1}$, and its domain is only $x\neq1$. The expression $\frac{x+1}{x-1}$ with $x\neq1$ has both the same simplified form and same domain. The other options either introduce extra excluded values or simplify to a different rational expression.
4. Which equation is true for all permitted values of x?
   Answer: $\frac{x^2-9}{x+3}=x-3$ When you see a rational equation with quadratic expressions, your first move should be to factor the numerator and denominator. This question is really testing whether you can recognize the difference of squares pattern: $x^2-9=(x-3)(x+3)$. Applying that to $\frac{x^2-9}{x+3}$, the numerator becomes $(x-3)(x+3)$, and the $x+3$ factors cancel, leaving $x-3$ — exactly the right side. Since division by zero is not permitted, this is true for all $x\neq -3$. Now check each trap. The choice $\frac{x^2+9}{x+3}=x+3$ is false because $(x+3)^2=x^2+6x+9$, not $x^2+9$; $x^2+9$ does not factor over real numbers. The choice $\frac{x^2-9}{x-3}=x-3$ has the correct factorization, but cancelling $x-3$ leaves $x+3$, not $x-3$. The choice $\frac{x^2+9}{x-3}=x+3$ also fails because the numerator is not $(x-3)(x+3)$; that would be $x^2-9$, so the equality cannot hold. Your strategy: whenever you see $x^2-a^2$, factor immediately into $(x-a)(x+a)$. Then simplify, and always note the excluded value that makes the denominator zero. On this exam, the most common wrong answer comes from confusing $x^2+9$ with a difference of squares — remember, a plus sign means you cannot factor it this way.
5. Simplify $\frac{\frac{x}{x-1}-\frac{x}{x+1}}{\frac{2x}{x^2-1}}$. Which of the following is the simplified expression with the correct domain restrictions?
   Answer: $1$, for $x\neq -1,0,1$ The numerator combines to $\frac{x(x+1)-x(x-1)}{x^2-1}=\frac{2x}{x^2-1}$. Dividing by $\frac{2x}{x^2-1}$ gives $1$. The original expression also requires $2x\neq0$, so $x=0$ is excluded in addition to $x\neq\pm1$. The $-1$ option is a sign error, and the remaining $1$ option omits the $x=0$ restriction.
6. Which expression has domain restrictions $x\neq -2,0,3$ and is equivalent to $\frac{x}{x+2}$ for all $x$ in its domain?
   Answer: $\frac{x^3-3x^2}{x^3-x^2-6x}$ The numerator of the correct expression factors as $x^2(x-3)$, and the denominator factors as $x(x-3)(x+2)$. Thus it simplifies to $\frac{x}{x+2}$ and is undefined at $x=0,3,-2$. The second option misses $x=0$, the third has different denominator factors, and the fourth simplifies to $\frac{x}{x-3}$ instead.
7. For what value of $a$ is $\frac{3}{x-a}+\frac{5}{x+a}=\frac{8x-4}{x^2-a^2}$ for all $x$ in the domain of the equation?
   Answer: $2$ Adding the left side gives $\frac{3(x+a)+5(x-a)}{x^2-a^2}=\frac{8x-2a}{x^2-a^2}$. Matching the numerator to $8x-4$ requires $-2a=-4$, so $a=2$. Choosing $-2$ is a sign error, $4$ gives the wrong constant term, and the identity is possible so the no-such-value option is false.
8. A company's revenue and cost from selling x items are $R=\frac{x^2-1}{x+2}$ and $C=\frac{x^2-2x-3}{x+2}$ thousand dollars. Which expression represents the profit $R-C$?
   Answer: $\frac{2(x+1)}{x+2}$ When you see profit as $R-C$, treat the two rational expressions as fractions with a common denominator, then subtract the numerators carefully. Here both have denominator $x+2$, so: $$R-C=\frac{x^2-1}{x+2}-\frac{x^2-2x-3}{x+2} =\frac{x^2-1-(x^2-2x-3)}{x+2}.$$ Now distribute the minus sign through the second numerator: $$x^2-1-x^2+2x+3=2x+2=2(x+1).$$ So the profit is $$\frac{2(x+1)}{x+2}.$$ This confirms the correct answer is the expression $\frac{2(x+1)}{x+2}$. Why the others miss the mark: - $\frac{2x+1}{x+2}$ looks close, but it loses the $2$ from combining $-1+3$. The numerator should be $2x+2$, not $2x+1$. - $\frac{-2(x+2)}{x+2}$ would simplify to $-2$, which suggests the subtraction sign was distributed incorrectly or the original numerators were combined with wrong signs. - $\frac{2(x-1)}{x+2}$ would come from getting $2x-2$ instead of $2x+2$, likely by treating $-1+3$ as $-2$ or factoring incorrectly. Your strategy: whenever subtracting rational expressions, put the entire second numerator in parentheses, then distribute the negative sign before combining like terms. This one small step prevents most sign and simplification errors on this type of exam question.
9. Which expression is equivalent to $\frac{x^2-5x+6}{x^2-4}\cdot \frac{x^2-x-2}{x^2-6x+9}\div \frac{x^2-1}{x^2-3x}$ and gives all of the domain restrictions?
   Answer: $\frac{x(x-2)}{(x+2)(x-1)}$; $x\neq -2,-1,0,1,2,3$ Factoring and using the reciprocal of the divisor gives $\frac{x(x-2)}{(x+2)(x-1)}$. The original denominators require $x\neq-2,2,0,3$, and the divisor is zero at $x=\pm1$, so the complete domain excludes $-2,-1,0,1,2,3$. The option missing $-1,1$ fails to account for dividing by zero, the option missing $0,3$ ignores the divisor denominator, and $\frac{x-2}{x+2}$ is an incorrect cancellation.
10. Which set lists all values that must be excluded from the domain of $\frac{x+2}{x^2+2x-3}+\frac{x-1}{x^2+5x+6}$?
    Answer: $x\neq -3,-2,1$ When you see a rational expression and the question asks for excluded domain values, your job is to find every number that would make any denominator zero. The numerators do not affect the domain—they only affect where the expression equals zero. So factor each denominator and solve. . The first denominator is $x^2+2x-3=(x+3)(x-1)$, so it is zero when $x=-3$ or $x=1$. The second denominator is $x^2+5x+6=(x+2)(x+3)$, so it is zero when $x=-2$ or $x=-3$. Combining these unique values gives $x\neq -3,-2,1$, which is the correct set. The choice listing $x\neq -3,-1,2$ includes the extraneous values $-1$ and $2$, omits $-2$ and $1$, and likely comes from mis-factoring the denominators or using numerator zeros. The choice listing $x\neq -1,1,2$ also inserts $-1$ and $2$ and omits both $-3$ and $-2$, so it misses two real restrictions while adding values that do not make any denominator zero. The choice listing $x\neq -3,-2$ correctly handles the second denominator, but forgets that the first denominator also excludes $x=1$. The correct answer is the one that includes all three restrictions, not the ones with $-1$ or $2$ thrown in. As a study tip: always factor both denominators completely before you list domain restrictions, and combine duplicate factors only once. A quick way to check yourself is to plug each candidate value into the original denominators; if either denominator becomes zero, the value must stay excluded.
11. After simplifying $\frac{x^2-4}{x^2-3x+2}\div\frac{x+2}{x^2-1}$, which choice gives the simplified expression and all excluded values from the original expression?
    Answer: $x+1$, $x\neq -2,-1,1,2$ When you see a division of rational expressions, factor everything first, but record excluded values before you cancel. Start with the original expression: $$\frac{x^2-4}{x^2-3x+2}\div\frac{x+2}{x^2-1} = \frac{(x-2)(x+2)}{(x-1)(x-2)}\cdot\frac{(x-1)(x+1)}{x+2}.$$ Before canceling, find every value that makes the original expression undefined. The first denominator gives $x\neq 1,2$. The second denominator $x^2-1$ gives $x\neq -1,1$. Also, the divisor is zero when $x+2=0$, so $x\neq -2$. Thus the complete restriction set is $$x\neq -2,-1,1,2.$$ Canceling the common factors $(x-2)$, $(x+2)$, and $(x-1)$ leaves **$x+1$**. So the simplified expression is $x+1$, with all four excluded values. The choice that lists only $x\neq -1,1,2$ misses $-2$; you cannot cancel the $x+2$ and ignore that it made the divisor zero. The choice that lists $x\neq -2,-1,2$ misses $1$; even though $x-1$ cancels, $x=1$ vas an original denominator zero. The choice that lists $x\neq -2,-1,1$ misses $2$; the first denominator $(x-1)(x-2)$ clearly excludes $x=2$. The complete version—$x+1$, $x\neq -2,-1,1,2$—alone accounts for all restrictions from the original expression. Strategy: always write domain restrictions from the original denominators and the divisor's numerator before simplifying. Then cancel factors for the simplified form, but keep the original restrictions.
12. How many real numbers must be excluded from the domain of $\frac{3x+6}{x^2+5x+6}\div \frac{x^2-4}{x^2+6x+9}$?
    Answer: $3$ The first denominator factors as $(x+2)(x+3)$, so $x\neq-2,-3$. The divisor denominator is $(x+3)^2$, so $x\neq-3$, and the divisor numerator is zero at $x=2,-2$, so those values also must be excluded. The distinct excluded values are $-3,-2,2$, giving 3 values. Counting only the simplified denominator misses the original division-by-zero restriction.
13. Which expression and domain restriction set correctly describe $\frac{\frac{2}{x+1}-1}{\frac{2}{x+1}+1}$?
    Answer: $\frac{1-x}{x+3}$, $x\neq -3,-1$ Whenever you simplify a rational expression, you must keep the domain restrictions from the original expression, not only from the simplified form. Start by multiplying the numerator and denominator by $x+1$: $\frac{\frac{2}{x+1}-1}{\frac{2}{x+1}+1} \cdot \frac{x+1}{x+1} = \frac{2-(x+1)}{2+(x+1)} = \frac{1-x}{x+3}.$ This simplification is valid only when $x+1\neq 0$, so $x\neq -1$ must be kept. Also, the simplified denominator $x+3$ cannot be zero, so $x\neq -3$. Thus the expression is $\frac{1-x}{x+3}$ with domain $x\neq -3,-1$. The choice $\frac{1-x}{x+3}$ with only $x\neq -3$ misses the original restriction $x\neq -1$. The choice $\frac{1-x}{x+3}$ with only $x\neq -1$ ignores that $x=-3$ makes the denominator zero after simplifying. And the choice $\frac{x-1}{x+3}$ with $x\neq -3,-1$ has the sign wrong: distributing the negative gives $2-(x+1)=2-x-1=1-x$, not $x-1$. A good habit: before simplifying, identify every value that makes any denominator zero; after simplifying, check the new denominator too. Combining both sets of restrictions gives the full domain.
14. A student simplifies $\frac{\frac{x^2-4}{x^2-2x}}{\frac{x+2}{x-3}}$ and obtains $\frac{x-3}{x}$. Which statement is correct?
    Answer: The simplification is correct, and the original expression is undefined for $x=-2,0,2,3$. The inner numerator factors to $\frac{(x+2)(x-2)}{x(x-2)}=\frac{x+2}{x}$. Dividing by $\frac{x+2}{x-3}$ gives $\frac{x-3}{x}$. The original expression is undefined at $x=0,2$ from $x^2-2x$, at $x=3$ from the divisor denominator, and at $x=-2$ because the divisor is zero. Thus the domain is $x\neq-2,0,2,3$.
15. For the rational expression $\frac{x^2-9}{x^2+x-6}$, which of the following correctly lists all values that must be excluded from the domain?
    Answer: $x\neq -3$ and $x\neq 2$ Whenever you see a rational expression and are asked about excluded values, your job is to find where the denominator equals zero — those are the values that make the expression undefined. Don't focus on the numerator; it affects zeros, not domain. Start by factoring the denominator: $x^2+x-6=(x+3)(x-2)$. Setting each factor equal to zero gives $x+3=0 \Rightarrow x=-3$ and $x-2=0 \Rightarrow x=2$. Therefore, both $x=-3$ and $x=2$ must be excluded. Notice that the numerator $x^2-9=(x-3)(x+3)$ shares a factor of $x+3$, but even though the expression simplifies to $\frac{x-3}{x-2}$, the original denominator still becomes zero at $x=-3$, so that value cannot enter the domain. Now look at the incorrect choices. "$x\neq -3$ only" misses $x=2$, so it excludes too few values. "$x\neq 3$ and $x\neq -2$" uses the factors from the numerator and the wrong signs; these are not denominator roots. "$x\neq 2$ only" falls for the common trap of simplifying first and forgetting that $x=-3$ is still forbidden in the original expression. Your strategy: always factor the original denominator before simplifying, then set each factor to zero. List every value that makes the original denominator zero — cancellation does not remove domain restrictions. This habit will save you on every rational-expression domain question.
16. Which choice gives the correct simplified form and the domain restrictions for the original expression $\frac{x^2+4x+4}{x^2+3x+2}$?
    Answer: $\frac{x+2}{x+1}$, $x\neq -1,-2$ Whenever you see a rational expression like this, your job is to factor both the numerator and denominator completely, then cancel only common factors. But the domain restrictions must come from the original denominator, before canceling. Factor the numerator: $x^2+4x+4 = (x+2)^2$. Factor the denominator: $x^2+3x+2 = (x+2)(x+1)$. So the expression becomes $\frac{(x+2)^2}{(x+2)(x+1)}$. Cancel one $x+2$ factor, leaving $\frac{x+2}{x+1}$. Now, the original denominator is zero when $x = -2$ or $x = -1$. Even though $x+2$ cancels, the original expression is undefined at $x=-2$, so the domain must exclude both values: $x \neq -1, -2$. That is the correct pairing. The choice giving $\frac{x+2}{x+1}$ but only stating $x \neq -1$ is a trap: it shows the correct simplification but forgets that a canceled factor still creates a domain restriction in the original expression. The choice with $\frac{x-2}{x+1}$ has a sign error in the numerator — it should be $x+2$, not $x-2$. The choice with $\frac{x+2}{x-1}$ incorrectly factored the denominator as $(x-1)(x+2)$, but the constant term in $x^2+3x+2$ is positive, so the factors must be $(x+1)(x+2)$. On exam day, factor everything first, then cancel, and always write domain restrictions from the original denominator. Watch for sign errors and for canceled factors that still restrict the domain.
17. Which choice gives the fully simplified expression and all values that must be excluded from the domain of the original expression? $\frac{x^2-1}{x^2+5x+6}\cdot\frac{x^2+2x-3}{x^2-4x+3}$
    Answer: $\frac{x^2-1}{x^2-x-6}$, $x\neq -3,-2,1,3$ When you see a product of rational expressions, factor every numerator and denominator before canceling; domain restrictions come from the original denominators, not the simplified result. Factoring gives $$\frac{(x-1)(x+1)}{(x+2)(x+3)}\cdot\frac{(x+3)(x-1)}{(x-3)(x-1)}.$$ Cancel the common factor $x+3$ and one $x-1$, leaving $$\frac{(x-1)(x+1)}{(x+2)(x-3)} = \frac{x^2-1}{x^2-x-6}.$$ Now set each original denominator factor to zero: $x+2=0$, $x+3=0$, $x-3=0$, and $x-1=0$. Therefore $x\neq -3,-2,1,3$. Notice that $x=-3$ and $x=1$ are excluded even though they cancel; the original expression is undefined there. So the correct choice is the one with $\frac{x^2-1}{x^2-x-6}$ and $x\neq -3,-2,1,3$. The choices listing $x\neq -2,3$ only or $x\neq -3,-2$ use the simplified denominator alone and ignore restrictions from the original second denominator. The choice listing $x\neq -3,-2,3$ catches most values but misses $x=1$, a classic trap because $x-1$ cancels. Study tip: after simplifying any rational expression, pause and list all zeros of the original denominators before writing the final domain. That habit prevents the most common domain-error trap on this exam.
18. Two resistors with resistances $x$ ohms and $x+4$ ohms are connected in parallel. The total resistance $R_T$ satisfies $\frac{1}{R_T}=\frac{1}{x}+\frac{1}{x+4}$. Which formula for $R_T$ is correct for the rational expression and gives its domain restrictions?
    Answer: $R_T=\frac{x(x+4)}{2(x+2)}$; $x\neq 0,-4,-2$ Adding the reciprocals gives $\frac{x+4+x}{x(x+4)}=\frac{2(x+2)}{x(x+4)}$. Taking the reciprocal gives $R_T=\frac{x(x+4)}{2(x+2)}$. The original sum excludes $x=0,-4$, and the denominator of the simplified form requires $x\neq-2$. The option missing $-2$ ignores the zero denominator after combining.
19. Which expression is equivalent to $\frac{x^2+2x+1}{x^2-1}\div\frac{x^2+x}{x^2-x}$ for all permitted values of x?
    Answer: $1$, $x\neq -1,0,1$ Whenever you see division of rational expressions, rewrite it as multiplication by the reciprocal, then factor and cancel. Start by factoring: $\frac{x^2+2x+1}{x^2-1}\div\frac{x^2+x}{x^2-x} = \frac{(x+1)^2}{(x-1)(x+1)}\cdot\frac{x(x-1)}{x(x+1)}.$ Now cancel matching factors: one $x+1$, one $x-1$, and the $x$. Everything cancels, leaving the multiplicative identity, $1$. Since no denominator can be zero, $x\neq -1,0,1$, so the equivalent expression is $1$ with those restrictions. The choice $\frac{x+1}{x-1}$ is what you get if you simplify only the first fraction and stop; it ignores the division. The choice $0$ is the "everything canceled, so nothing remains" misconception — canceling factors leaves $1$, not $0$. The choice $\frac{x-1}{x+1}$ is the reciprocal of the simplified second fraction; it appears if you flip a fraction but forget to multiply by the first fraction. Only the full simplification gives $1$. Study tip: always write division as "multiply by the reciprocal" before canceling. Canceling across a division sign is a classic error. And if all factors cancel, the result is $1$, not $0$.
20. A rectangle has area $\frac{2x^2+3x-2}{x^2-9}$ square units and width $\frac{x+2}{x^2-5x+6}$ units. Which expression represents the length of the rectangle, assuming no denominator is zero?
    Answer: $\frac{(2x-1)(x-2)}{x+3}$ Whenever a rectangle problem gives you area and width as rational expressions, remember that length is area divided by width. So you should multiply the area by the reciprocal of the width, then factor everything before canceling. Here: $$\frac{2x^2+3x-2}{x^2-9}\div\frac{x+2}{x^2-5x+6} = \frac{(2x-1)(x+2)}{(x-3)(x+3)} \cdot \frac{(x-2)(x-3)}{x+2}.$$ Cancel the common $(x+2)$ and $(x-3)$ factors, and you are left with $$\frac{(2x-1)(x-2)}{x+3}.$$ The expression with $x-3$ in the denominator comes from incomplete canceling: the $x-3$ factor must cancel, leaving $x+3$ below. The one with $x+2$ in the numerator usually means $x^2-5x+6$ was mis-factored as $(x-3)(x+2)$, so the $x+2$ never gets canceled. The one with $x-3$ in the numerator keeps the wrong factor from $x^2-5x+6$; the correct factor to keep after canceling is $x-2$, not $x-3$. Only $\frac{(2x-1)(x-2)}{x+3}$ has the correct factors after full simplification. Strategy: factor every numerator and denominator completely, rewrite division as multiplication by the reciprocal, and only cancel binomials that appear in both a numerator and a denominator.

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