Question

In Exercises 33–68, add or subtract as indicated. (4x2+x−6)/(x2+3x+2)−3x/(x+1)+5/(x+2)

Accepted Answer

Factorize the denominators where possible. For the first term, $$x^2 + 3x + 2$$ can be factored as $$(x + 1)(x + 2). $$The other denominators $$x + 1$$ and $$x + 2$$ are already in their simplest forms. Identify the least common denominator (LCD). The LCD for the denominators $$(x + 1)(x + 2)$$, $$x + 1$$, and $$x + 2 is$$ $$(x + 1)(x + 2). $$Rewrite each term with the LCD as the denominator. For the first term, it already has $$(x + 1)(x + 2) as $$the denominator. For the second term, multiply both numerator and denominator by $$x + 2$$, resulting in $$\frac{3x(x + 2)}{(x + 1)(x + 2)}. $$For the third term, multiply both numerator and denominator by $$x + 1$$, resulting in $$\frac{5(x + 1)}{(x + 1)(x + 2)}. $$Combine all terms into a single fraction with the common denominator $$(x + 1)(x + 2). $$The numerator becomes $$(4x^2 + x - 6) - 3x(x + 2) + 5(x + 1). $$Simplify the numerator by distributing and combining like terms. Expand $$-3x(x + 2)$$ and $$5(x + 1)$$, then combine all terms in the numerator. The final expression will be a single fraction with the simplified numerator over $$(x + 1)(x + 2).$$

In Exercises 33–68, add or subtract as indicated. (4x²+x−6)/(x²+3x+2)−3x/(x+1)+5/(x+2)

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Key Concepts

Rational Expressions

Finding a Common Denominator

Polynomial Factoring