Simplifying Rational Expressions

Section 7.2 Simplifying Rational Expressions

Recall that simplifying a fraction involves cancelling the greatest common factor from the numerator and the denominator, so the fraction is in lowest terms.

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Example 7.2.1. Simplifying a Numerical Fraction.

To simplify $\dfrac{36}{60}\text{,}$ factor both the numerator and denominator,

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\begin{align*} \dfrac{36}{60} \amp = \dfrac{2 \times 2 \times 3 \times 3}{2 \times 2 \times 3 \times 5} \end{align*}

Cancel the common factors from the numerator and denominator,

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\begin{align*} \dfrac{\cancel{2} \times \cancel{2} \times \cancel{3} \times 3}{\cancel{2} \times \cancel{2} \times \cancel{3} \times 5} \amp = \dfrac{3}{5} \end{align*}

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Similarly, simplifying a rational expression involves factoring the numerator and denominator, and cancelling common factors until there are no common factors other than 1. In this case, we say the rational expression is in lowest terms.

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Subsection 7.2.1 Simplifying Expressions with Monomials

Exercise Group 7.2.1. Monomial Expressions.

Simplify each rational expression, and state any restrictions on the variables.

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(a)

$\dfrac{12x}{48x^3}$

Answer.

$\dfrac{1}{4x^2}, x\neq 0$

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(b)

$\dfrac{21ab}{35ab^2}$

Answer.

$\dfrac{3}{5b}, a\neq 0, b\neq 0$

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(c)

$\dfrac{8m^2n}{24mn^3}$

Answer.

$\dfrac{m}{3n^2}, m\neq 0, n\neq 0$

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(d)

$\dfrac{16x^3y^3}{36x^5y^2}$

Answer.

$\dfrac{4y}{9x^2}, x\neq 0, y\neq 0$

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(e)

$\dfrac{12a^2b^3}{40a^3b}$

Answer.

$\dfrac{3b^2}{10a}, a\neq 0, b\neq 0$

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(f)

$\dfrac{6xyz^3}{3x^2y^2z}$

Answer.

$\dfrac{2z^2}{xy}, x\neq 0, y\neq 0, z\neq 0$

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(g)

$\dfrac{8x}{14y}$

Answer.

$\dfrac{4x}{7y}, y\neq 0$

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(h)

$\dfrac{15xy}{24x}$

Answer.

$\dfrac{5y}{8}, x\neq 0$

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(i)

$\dfrac{12ab}{6a^2b^2}$

Answer.

$\dfrac{2}{ab}, a\neq 0, b\neq 0$

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(j)

$\dfrac{7x^3}{21x^8}$

Answer.

$\dfrac{1}{3x^5}, x\neq 0$

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(k)

$\dfrac{4x^2yz^5}{6x^2y^3z^2}$

Answer.

$\dfrac{2z^3}{3y^2}, x\neq 0, y\neq 0, z\neq 0$

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(l)

$\dfrac{30x^4y^3}{-6x^7y}$

Answer.

$-\dfrac{5y^2}{x^3}, x\neq 0, y\neq 0$

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(m)

$\dfrac{9x^2}{6x^3}$

Answer.

$\dfrac{3}{2x}, x\neq 0$

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(n)

$\dfrac{7a^2b^3}{21a^4b}$

Answer.

$\dfrac{b^2}{3a^2}, a\neq 0, b\neq 0$

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Sometimes, factors can have more than one term in them.

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Exercise Group 7.2.2. Binomial Factors.

Simplify each rational expression, and state any restrictions on the variables.

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(a)

$\dfrac{6(x+2)}{10(x+2)}$

Answer.

$\dfrac{3}{5}, x\neq -2$

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(b)

$\dfrac{(x+1)(x-1)}{(x-1)^2}$

Answer.

$\dfrac{x+1}{x-1}, x\neq 1$

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(c)

$\dfrac{5(x+3)}{(x+3)(x-3)}$

Answer.

$\dfrac{5}{x-3}, x\neq -3,3$

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(d)

$\dfrac{(x-1)(x-3)}{(x+2)(x-1)}$

Answer.

$\dfrac{x-3}{x+2}, x\neq -2,1$

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Subsection 7.2.2 Simplifying Expressions with Polynomials by Factoring

Look for common factors.

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Exercise Group 7.2.3. Polynomial Expressions.

Simplify each rational expression, and state any restrictions on the variables.

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(a)

$\dfrac{6-4x}{2}$

Answer.

$3-2x\text{,}$ none

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(b)

$\dfrac{3x-3}{6x-6}$

Answer.

$\dfrac{1}{2}, x\neq 1$

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(c)

$\dfrac{4x-4}{4x+4}$

Answer.

$\dfrac{x-1}{x+1}, x\neq -1$

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(d)

$\dfrac{6x-9}{2x-3}$

Answer.

$3, x\neq \dfrac{3}{2}$

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(e)

$\dfrac{x^2 - 4}{2x - 4}$

Answer.

$\dfrac{x + 2}{2}, x\neq 2$

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(f)

$\dfrac{x+6}{3x+18}$

Answer.

$\dfrac{1}{3}, x\neq -6$

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(g)

$\dfrac{x^2-64}{x+8}$

Answer.

$x-8, x\neq -8$

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(h)

$\dfrac{x+1}{x^2-1}$

Answer.

$\dfrac{1}{x-1}, x\neq -1,1$

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(i)

$\dfrac{x^2-7x-18}{x-9}$

Answer.

$x+2, x\neq 9$

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(j)

$\dfrac{x-2}{x^2-4}$

Answer.

$\dfrac{1}{x+2}, x\neq -2,2$

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(k)

$\dfrac{x+6}{x^2+2x-24}$

Answer.

$\dfrac{1}{x-4}, x\neq -6,4$

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(l)

$\dfrac{2x+10}{x^2-2x-35}$

Answer.

$\dfrac{2}{x-7}, x\neq -5,7$

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(m)

$\dfrac{x^2+9x-10}{3x+30}$

Answer.

$\dfrac{x-1}{3}, x\neq -10$

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(n)

$\dfrac{6x^3-8x}{9x^3-12x}$

Answer.

$\dfrac{2}{3}, x\neq 0, \pm\dfrac{2\sqrt{3}}{3}$

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(o)

$\dfrac{8x^3+8x^2}{10x^2+10x}$

Answer.

$\dfrac{4x}{5}, x\neq -1,0$

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Sometimes, two factors will look similar but not be exactly the same. In particular, they will be negatives of each other.

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Example 7.2.2. Factors Differing by a Negative.

For example, consider,

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\begin{gather*} \dfrac{2-x}{x-2} \end{gather*}

The negative of $2-x$ is $-(2-x) = -2+x = x-2\text{.}$ If you divide a number and its negative (like e.g. 5 and $-5$), the result is always $-1\text{.}$ We can factor out a $-1$ from the numerator, and then cancel,

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\begin{align*} \amp = \dfrac{-(x-2)}{x-2} = \dfrac{-1}{1} = -1 \end{align*}

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Exercise Group 7.2.4. Simplifying Negative Factors.

Simplify each rational expression.

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(a)

$\dfrac{x-1}{x-1}$

Answer.

$1$

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(b)

$\dfrac{x-1}{1-x}$

Answer.

$-1$

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(c)

$\dfrac{x+1}{1+x}$

Answer.

$1$

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(d)

$\dfrac{x+2}{-2-x}$

Answer.

$-1$

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(e)

$\dfrac{(x-1)(x+1)}{(1-x)(-x-1)}$

Answer.

$1$

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(f)

$\dfrac{3-x}{x-3}$

Answer.

$-1$

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(g)

$\dfrac{x^2-4}{4-x^2}$

Answer.

$-1$

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(h)

$\dfrac{x^2-2x+1}{-x^2+2x-1}$

Answer.

$-1$

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(i)

$\dfrac{5 - x}{x - 5}$

Answer.

$-1$

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(j)

$\dfrac{(x-2)(x-1)(x+2)}{(2-x)(1-x)(2-x)}$

Answer.

$-\dfrac{x+2}{x-2}$ or $\dfrac{-x-2}{x-2}$

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(k)

$\dfrac{2x(3-x)}{5x^2(x-3)}$

Answer.

$-\dfrac{2}{5x}$

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(l)

$\dfrac{a-b+c}{b-c-a}$

Answer.

$-1$

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Exercise Group 7.2.5. Simplifying and Restrictions.

Simplify each rational expression, and state any restrictions on the variables.

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(a)

$\dfrac{2x^2-8x}{4-x}$

Answer.

$-2x, x\neq 4$

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(b)

$\dfrac{20x^3+15x^2-5x}{5x}$

Answer.

$4x^2+3x-1, x\neq 0$

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(c)

$\dfrac{3x-21}{28-4x}$

Answer.

$-\dfrac{3}{4}, x\neq 7$

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(d)

$\dfrac{x+2}{x^2-4}$

Answer.

$\dfrac{1}{x-2}, x\neq -2,2$

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(e)

$\dfrac{2x-8}{x^2-16}$

Answer.

$\dfrac{2}{x+4}, x\neq -4,4$

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(f)

$\dfrac{8x^3-10x}{20x^3-25x}$

Answer.

$\dfrac{2}{5}, x\neq 0, \pm\dfrac{\sqrt{5}}{2}$

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(g)

$\dfrac{5(4x-2)}{8(2x-1)^2}$

Answer.

$\dfrac{5}{4(2x-1)}, x\neq \dfrac{1}{2}$

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(h)

$\dfrac{10x^4-8x^2+4x}{2x^2}$

Answer.

$\dfrac{5x^3-4x+2}{x}, x\neq 0$

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Exercise Group 7.2.6. Quadratic Expressions.

Simplify each rational expression.

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(a)

$\dfrac{x^2 + 4x - 12}{3x - 6}$

Answer.

$\dfrac{x + 6}{3}$

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(b)

$\dfrac{12 + x - x^2}{2x^2 - 9x + 4}$

Answer.

$\dfrac{-(x+3)}{2x-1}$ or $\dfrac{-3 - x}{2x - 1}$

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(c)

$\dfrac{x^2 - 6x + 8}{5x - 20}$

Answer.

$\dfrac{x - 2}{5}$

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(d)

$\dfrac{x^2 + 2x - 24}{2x^2 - 72}$

Answer.

$\dfrac{x-4}{2(x-6)}$

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(e)

$\dfrac{x^2-x-6}{x^2-4}$

Answer.

$\dfrac{x-3}{x-2}$

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(f)

$\dfrac{x^2-9}{12-7x+x^2}$

Answer.

$\dfrac{x+3}{x-4}$

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(g)

$\dfrac{x^2+5x+6}{x^2+6x+8}$

Answer.

$\dfrac{x+3}{x+4}$

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(h)

$\dfrac{x^2+3x-4}{x^2+2x-8}$

Answer.

$\dfrac{x-1}{x-2}$

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(i)

$\dfrac{4x^2-20x}{x^2-4x-5}$

Answer.

$\dfrac{4x}{x+1}$

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(j)

$\dfrac{5x^2+10x-40}{10x^2-30x+20}$

Answer.

$\dfrac{x+4}{2(x-1)}$

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(k)

$\dfrac{4x-20}{3x^2-14x-5}$

Answer.

$\dfrac{4}{3x+1}$

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(l)

$\dfrac{2x^2+12x+18}{3x^2-3x-36}$

Answer.

$\dfrac{2(x+3)}{3(x-4)}$

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(m)

$\dfrac{4x^2+16x}{x^2-16}$

Answer.

$\dfrac{4x}{x-4}$

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(n)

$\dfrac{x^2+2x}{x^2+3x+2}$

Answer.

$\dfrac{x}{x+1}$

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(o)

$\dfrac{x^2+9x+18}{x^2+6x}$

Answer.

$\dfrac{x+3}{x}$

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(p)

$\dfrac{x^2+7x-18}{12-4x-x^2}$

Answer.

$-\dfrac{x+9}{x+6}$

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(q)

$\dfrac{5x-10}{x^2+5x-14}$

Answer.

$\dfrac{5}{x+7}$

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(r)

$\dfrac{x^2+2x+1}{x^2-1}$

Answer.

$\dfrac{x+1}{x-1}$

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(s)

$\dfrac{x^2-25}{x^2-4x-5}$

Answer.

$\dfrac{x+5}{x+1}$

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(t)

$\dfrac{x^2+6x-16}{x^2-11x+18}$

Answer.

$\dfrac{x+8}{x-9}$

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(u)

$\dfrac{x^2+x-6}{2x^2-8}$

Answer.

$\dfrac{x+3}{2(x+2)}$

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(v)

$\dfrac{x^2-2x+1}{2x^2-x-1}$

Answer.

$\dfrac{x-1}{2x+1}$

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(w)

$\dfrac{2x^2-4x-6}{2x^2-8x-10}$

Answer.

$\dfrac{x-3}{x-5}$

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(x)

$\dfrac{x^2+7x-8}{2-2x}$

Answer.

$-\dfrac{x+8}{2}$

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Exercise Group 7.2.7. Complex Polynomial Expressions.

Simplify each rational expression.

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(a)

$\dfrac{x^2-9x+20}{16-x^2}$

Answer.

$-\dfrac{x-5}{x+4}$

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(b)

$\dfrac{x^2 - 16}{2x^2 + 7x - 4}$

Answer.

$\dfrac{x - 4}{2x - 1}$

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(c)

$\dfrac{6x^2+5x-4}{4x^2-1}$

Answer.

$\dfrac{3x+4}{2x+1}$

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(d)

$\dfrac{2x^2+9x+4}{4x^2-4x-3}$

Answer.

$\dfrac{x+4}{2x-3}$

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(e)

$\dfrac{2x^2+5x-3}{x^2-9}$

Answer.

$\dfrac{2x-1}{x-3}$

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(f)

$\dfrac{3x^2-10x-8}{x^2-16}$

Answer.

$\dfrac{3x+2}{x+4}$

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(g)

$\dfrac{2x^2+17x+35}{3x^2+19x+20}$

Answer.

$\dfrac{2x+7}{3x+4}$

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(h)

$\dfrac{5x^2-32x+12}{4x^2-27x+18}$

Answer.

$\dfrac{5x-2}{4x-3}$

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(i)

$\dfrac{7x^2+61x-18}{7x^2+19x-6}$

Answer.

$\dfrac{x+9}{x+3}$

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(j)

$\dfrac{8x^2-51x+18}{8x^2+29x-12}$

Answer.

$\dfrac{x-6}{x+4}$

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(k)

$\dfrac{5x^2-9x-2}{x^2+4x-12}$

Answer.

$\dfrac{5x+1}{x+6}$

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(l)

$\dfrac{4x^2-x-3}{4x^2-17x-15}$

Answer.

$\dfrac{x-1}{x-5}$

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(m)

$\dfrac{3x^3-3x^2}{8x^3-12x^2+4x}$

Answer.

$\dfrac{3x}{4(2x-1)}$

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Exercise Group 7.2.8. Trinomials and Restrictions.

Simplify each rational expression, and state any restrictions on the variables.

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(a)

$\dfrac{5x^2+x-4}{25x^2-40x+16}$

Answer.

$\dfrac{x+1}{5x-4}, x\neq\dfrac{4}{5}$

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(b)

$\dfrac{14x^3-7x^2+21x}{7x}$

Answer.

$2x^2-x+3, x\neq0$

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(c)

$\dfrac{2x^2+10x}{-3x-15}$

Answer.

$-\dfrac{2x}{3}, x\neq-5$

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(d)

$\dfrac{x+4}{x^2+3x-4}$

Answer.

$\dfrac{1}{x-1}, x\neq-4,1$

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(e)

$\dfrac{x^2-9}{15-5x}$

Answer.

$-\dfrac{x+3}{5}, x\neq3$

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(f)

$\dfrac{x^2-5x+6}{x^2+3x-10}$

Answer.

$\dfrac{x-3}{x+5}, x\neq-5,2$

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(g)

$\dfrac{10+3x-x^2}{25-x^2}$

Answer.

$\dfrac{x+2}{x+5}, x\neq-5,5$

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(h)

$\dfrac{x^2-7x+12}{x^3-6x^2+9x}$

Answer.

$\dfrac{x-4}{x(x-3)}, x\neq0,3$

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(i)

$\dfrac{6x^2-x-2}{2x^2-x-1}$

Answer.

$\dfrac{3x-2}{x-1}, x\neq-\dfrac{1}{2},1$

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Subsection 7.2.3 Common Mistake: Cancelling “Common Terms”

Common factors can be cancelled, but not “common terms”. For example,

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\begin{align*} \dfrac{x + 1}{x} \amp = \dfrac{\cancel{x} + 1}{\cancel{x}} \amp\amp \text{$x$ is not a factor of the numerator}\\ \dfrac{x^2 + x + 3}{x^2 + 3} \amp = \dfrac{\cancel{x^2} + x + \cancel{3}}{\cancel{x^2} + \cancel{3}} \amp\amp \text{$x^2$ and $3$ are terms, not factors, so they can't be cancelled}\\ \amp \amp\amp \text{(This expression is already in lowest terms)} \notag\\ \dfrac{x - 2}{x + 1} \amp = \dfrac{\cancel{x} - 2}{\cancel{x} + 1} \amp\amp \text{$x$ is a term, not a factor}\\ \dfrac{2x^2 - 9}{2x - 3} \amp = \dfrac{\cancelto{x}{2x^2} - \cancelto{3}{9}}{\cancel{2x} - \cancel{3}} \amp\amp \text{$2x^2$, $2x$, $9$, and $3$ are all terms, not factors}\\ \dfrac{x^2 + 2x + 1}{x^2 + 3} \amp = \dfrac{\cancel{x^2} + 2x + 1}{\cancel{x^2} + 3} \amp\amp \text{$x^2$ are both terms, not factors} \end{align*}

In other words, something must be a factor of the entire numerator in order to cancel them.

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Subsection 7.2.4 Summary of Simplifying Rational Expressions

Factor the numerator and denominator as much as possible (if there are multiple terms, with addition or subtraction).
- If there is multiplication only, there is no factoring needed.
  
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Cancel common factors.
- Remember: only factors can cancel (not terms, which are added or subtracted).
  
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- Sometimes, common factors will differ by a constant factor like $-1\text{.}$
  
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Rewrite the fraction, with what’s left after cancelling.

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Find restrictions (if needed). Find the values of the variable that make the denominator 0 (from the original denominator, before any cancellation).

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Subsection 7.2.5 Word Problems

Example 7.2.3. Isosceles Triangle.

An isosceles triangle has two sides of length $9x+3$ and a perimeter of $30x+10\text{,}$ determine the ratio of the base to the perimeter in simplified form and state the restriction on $x\text{,}$ then explain in one sentence why that restriction is necessary in this situation.

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Answer.

$\dfrac{b}{P}=\dfrac{12x+4}{30x+10}=\dfrac{2}{5}\text{,}$ restriction $x\gt-\dfrac{1}{3}$ because the side lengths and perimeter must be positive (at $x=-\dfrac{1}{3}$ they are zero).

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Example 7.2.4. Rectangle Ratio.

A rectangle is six times as long as it is wide, determine the ratio of its area to its perimeter in simplest form if its width is $w\text{.}$

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Answer.

$\dfrac{A}{P}=\dfrac{6w^2}{14w}=\dfrac{3w}{7}$

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Example 7.2.5. Equivalent Rational Expressions.

The quotient of two polynomials is $3x-2\text{,}$ give two examples of a rational expression equivalent to this polynomial that has the restriction $x\neq 4\text{.}$

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Answer.

$\dfrac{(3x-2)(x-4)}{x-4}$ and $\dfrac{(3x-2)(x-4)^2}{(x-4)^2}$ (each simplifies to $3x-2$ for $x\neq 4$).

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Example 7.2.6. Consecutive Restrictions.

Give an example of a rational expression that could have three restrictions that are consecutive numbers.

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Answer.

e.g. $\dfrac{x^2+1}{(x-2)(x-3)(x-4)}$ (restrictions $x\neq 2,3,4$ are consecutive).

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Example 7.2.7. Box Dimensions.

The volume of a box is given by $(6x^2+23x+21)(x+5)$ and its height is $2x+3\text{.}$ Find a simplified expression for the area of the bottom of the box.

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Answer.

$A=\dfrac{(6x^2+23x+21)(x+5)}{2x+3}=(3x+7)(x+5)$

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