This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic: for addition, find the LCD, rewrite with common denominator, then add numerators. To add 2/(x - 3) + 5/(x + 2), we need LCD = (x - 3)(x + 2). Rewrite each fraction: 2/(x - 3) = 2(x + 2)/[(x - 3)(x + 2)] = (2x + 4)/[(x - 3)(x + 2)] and 5/(x + 2) = 5(x - 3)/[(x - 3)(x + 2)] = (5x - 15)/[(x - 3)(x + 2)]. Now add numerators: (2x + 4 + 5x - 15)/[(x - 3)(x + 2)] = (7x - 11)/[(x - 3)(x + 2)]. Choice C correctly shows this result, which equals (7x - 11)/(x² - x - 6) since (x - 3)(x + 2) = x² - x - 6. Choice B incorrectly adds as (7x + 4)—a sign error when distributing 5(x - 3) = 5x - 15, not 5x + 15. The rational expression operation hierarchy: Addition and subtraction are harder—find LCD as the product of distinct factors, rewrite each fraction with LCD, then add/subtract numerators. Always expand and combine like terms carefully!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. The golden rule: factor everything BEFORE operating! To find the LCD of 3/(x² - 4) and 2/(x² + 4x + 4), factor the denominators: x² - 4 = (x - 2)(x + 2) and x² + 4x + 4 = (x + 2)². The LCD must contain all factors with their highest powers: (x - 2) appears once, and (x + 2) appears squared. Therefore, LCD = (x - 2)(x + 2)². Choice C correctly identifies this LCD in factored form. Choice A incorrectly gives just (x - 2)(x + 2), missing that (x + 2) appears squared in the second denominator—the LCD must include the highest power of each factor. The rational expression operation hierarchy: For addition/subtraction, factor denominators completely, then the LCD is the product of all distinct factors, each raised to its highest power appearing in any denominator. This systematic approach ensures you can rewrite all fractions with the same denominator!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! For this subtraction, factor x² - 4 = (x - 2)(x + 2) as the LCD; rewrite the second as 3(x + 2)/((x - 2)(x + 2)); subtract numerators (x + 1) - 3(x + 2) = x + 1 - 3x - 6 = -2x - 5, giving (-2x - 5)/((x - 2)(x + 2)) in lowest terms. Choice A correctly performs the subtraction and simplifies, giving (-2x - 5)/((x - 2)(x + 2)) in lowest terms. A common mistake is switching signs in subtraction, leading to positive terms like in choice C, but remember subtraction means minus the entire numerator. Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In [2/(x - 1)] + [3/(x - 1)], you can't cancel the (x - 1)—you add the numerators: (2 + 3)/(x - 1) = 5/(x - 1). But in [2/(x - 1)] · [3/(x - 1)], you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! This complex fraction simplifies by multiplying by the reciprocal: rac{x}{x - 2} cdot rac{x + 1}{3} = rac{x(x + 1)}{3(x - 2)}, with no common factors to cancel further. Choice B correctly simplifies the complex fraction to rac{x(x + 1)}{3(x - 2)}. Choice D fails by perhaps canceling incorrectly across the complex structure, leading to an oversimplified rac{x}{3}. Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In [2/(x - 1)] + [3/(x - 1)], you can't cancel the (x - 1)—you add the numerators: (2 + 3)/(x - 1) = 5/(x - 1). But in [2/(x - 1)] · [3/(x - 1)], you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! To add these, use LCD (x - 1)(x + 1) = x² - 1; rewrite as x(x + 1)/(x² - 1) + 2(x - 1)/(x² - 1); combine numerators x² + x + 2x - 2 = x² + 3x - 2, giving (x² + 3x - 2)/(x² - 1) in lowest terms. Choice B correctly finds the LCD and combines, giving (x² + 3x - 2)/(x² - 1) in lowest terms. A common mistake is incorrectly expanding numerators, like forgetting the -2, leading to something like choice C, but verify the combined terms. Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In [2/(x - 1)] + [3/(x - 1)], you can't cancel the (x - 1)—you add the numerators: (2 + 3)/(x - 1) = 5/(x - 1). But in [2/(x - 1)] · [3/(x - 1)], you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! To divide, multiply by reciprocal: factor $x^2-1=(x-1)(x+1)$, $x^2+4x+3=(x+1)(x+3)$, so $$\frac{(x-1)(x+1)}{(x+1)(x+3)} \cdot \frac{x+1}{x-1}$$; cancel $(x-1)$ and $(x+1)$ to get $$\frac{x+1}{x+3}$$, simplified. Choice A correctly performs the division by factoring, multiplying by the reciprocal, and canceling, giving $$\frac{x+1}{x+3}$$ in lowest terms. Choice B fails by not fully canceling or perhaps forgetting to flip the divisor—always use the reciprocal for division! Common mistake: trying to cancel in addition before getting common denominator. You can only cancel FACTORS (things multiplied), never TERMS (things added). In $$ \frac{2}{x - 1} + \frac{3}{x - 1} $$, you can't cancel the $(x - 1)$—you add the numerators: $$ \frac{2 + 3}{x - 1} = \frac{5}{x - 1} $$. But in $$ \frac{2}{x - 1} \cdot \frac{3}{x - 1} $$, you CAN simplify the result. Know when canceling is valid (multiplication/division after factoring) vs invalid (addition/subtraction without common denominator)!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for division, multiply by the reciprocal (flip the second fraction). The closure property guarantees your result is always another rational expression! To divide (x² - 4)/(x² - 1) by (x - 2)/(x + 1), we multiply by the reciprocal: [(x² - 4)/(x² - 1)] · [(x + 1)/(x - 2)]. Factor everything first: x² - 4 = (x - 2)(x + 2) and x² - 1 = (x - 1)(x + 1). This gives us [(x - 2)(x + 2)]/[(x - 1)(x + 1)] · [(x + 1)/(x - 2)]. Now cancel: (x - 2) cancels from numerator and denominator, and (x + 1) cancels from numerator and denominator, leaving (x + 2)/(x - 1). Choice A correctly performs the division by multiplying by the reciprocal, factors, and cancels to get the simplified result. Choice B incorrectly inverts the final answer, C shows an intermediate step without simplification, and D forgot to cancel the (x + 1) factor. Common mistake: forgetting to flip the second fraction when dividing. Remember: division means multiply by the reciprocal, then factor and cancel as usual!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. The golden rule: factor everything BEFORE operating! To divide (x² + 5x + 6)/(x² - 1) by (x + 2)/(x - 1), multiply by the reciprocal. First factor: x² + 5x + 6 = (x + 2)(x + 3) and x² - 1 = (x - 1)(x + 1). This gives [(x + 2)(x + 3)]/[(x - 1)(x + 1)] · [(x - 1)/(x + 2)]. Now (x + 2) cancels and (x - 1) cancels, leaving (x + 3)/(x + 1). Choice A correctly shows this simplified result. Choice B incorrectly shows (x + 3)/(x + 2)—perhaps forgetting to flip the second fraction before multiplying. The rational expression operation hierarchy: For division, always flip the second fraction and multiply. Factoring first prevents working with huge expressions and makes cancellation obvious!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. The golden rule: factor everything BEFORE operating! Multiplying (x² - 4)/(x + 3) by (x + 3)/(x + 2) looks messy, but factor x² - 4 = (x + 2)(x - 2) first: [(x + 2)(x - 2)/(x + 3)] · [(x + 3)/(x + 2)]. Now (x + 3) and (x + 2) cancel immediately, leaving just (x - 2). Factoring first prevents working with huge expressions! To find the LCD, factor denominators: x²-4 = (x-2)(x+2), x²+4x+4 = $(x+2)^2$; the least common multiple takes the highest powers, so $(x-2)(x+2)^2$. Choice C correctly factors the denominators and takes the LCM with highest powers, giving $(x-2)(x+2)^2$. Choice B fails by not using the highest power of (x+2), perhaps overlooking the square—always check exponents in factoring! The rational expression operation hierarchy: Multiplication and division are easier—factor everything, cancel common factors, then multiply (or flip and multiply for division). Addition and subtraction are harder—factor denominators, find LCD as the LCM of denominator factors, rewrite each fraction with LCD, then add/subtract numerators. Always simplify at the end by factoring the numerator and canceling any new common factors with the denominator. This systematic approach prevents errors!

This question tests your understanding that rational expressions (fractions with polynomials) form a system just like rational numbers—closed under addition, subtraction, multiplication, and division by nonzero expressions. Operating on rational expressions uses the same rules as fraction arithmetic, just with polynomials instead of integers: for multiplication, multiply numerators and denominators (but factor and cancel first to keep things simple!); for division, multiply by the reciprocal (flip the second fraction); for addition/subtraction, find the LCD, rewrite with common denominator, then add/subtract numerators. The closure property guarantees your result is always another rational expression! For this subtraction, use LCD (x - 2)(x + 2) = x² - 4; the first is 3/(x² - 4), second is 1/(x - 2) = (x + 2)/(x² - 4); subtract numerators 3 - (x + 2) = 1 - x, giving (1 - x)/(x² - 4) or equivalently (-x + 1)/((x - 2)(x + 2)) in lowest terms. Choice C correctly performs the subtraction and simplifies, giving (-x + 1)/((x - 2)(x + 2)) in lowest terms. A common mistake is forgetting the negative in subtraction, leading to positive numerators like in choice D, but apply the minus to the entire second numerator. The rational expression operation hierarchy: Multiplication and division are easier—factor everything, cancel common factors, then multiply (or flip and multiply for division). Addition and subtraction are harder—factor denominators, find LCD as the LCM of denominator factors, rewrite each fraction with LCD, then add/subtract numerators. Always simplify at the end by factoring the numerator and canceling any new common factors with the denominator. This systematic approach prevents errors!