15. Simplifying Complicated Expressions

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Learning Objective:

Simplify a rational expression to determine what happens to the expression as its denominator goes to zero. Simplify a rational expression to determine when the expression evaluates to zero.

Video:
Interactive Activity

PDF Notes Link:

Simplifying Complicated ExpressionsDownload

Notes:

Simplifying expressions to allow us to determine what happens to the expression as ℎ → 0

Example 1

\frac{(x+h)^{2}-x^{2}}{h}=\frac{x^{2}+2xh+h^{2}-x^{2}}{h}=\frac{2xh+h^{2}}{h}=\frac{h(2x+h)}{h}=2x+h,\{\forall h\ne0\}

\text{This expression } \rightarrow 2x \text{ as } h \rightarrow 0.

Example 2

\frac{\sqrt{x+h}-\sqrt{x}}{h}=\frac{\sqrt{x+h}-\sqrt{x}}{h}(\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}})=\frac{h}{h(\sqrt{x+h}+\sqrt{x})}

=\frac{1}{\sqrt{x+h}+\sqrt{x}},\{\forall h\ne0\} \text{ (note use of the conjugate).}

\text{This expression } \rightarrow \frac{1}{2\sqrt{x}} \text{ as } h \rightarrow 0.

Example 3

\frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\frac{\frac{1}{x+h}(\frac{x}{x})-(\frac{x+h}{x+h})\frac{1}{x}}{h}=\frac{\frac{x-(x+h)}{(x+h)x}}{h}=\frac{-h}{h(x+h)x}=-\frac{1}{(x+h)x},\{\forall h\ne0\}.

\text{This expression } \rightarrow -\frac{1}{x^{2}} \text{ as } h \rightarrow 0.

Example 4

\frac{\frac{3}{x+2}-\frac{3}{a+2}}{x-a}=\frac{\frac{3}{x+2}(\frac{a+2}{a+2})-(\frac{x+2}{x+2})\frac{3}{a+2}}{x-a}=\frac{\frac{3a+6-(3x+6)}{(x+2)(a+2)}}{x-a}=\frac{3(a-x)}{(x-a)(x+2)(a+2)}

=-\frac{3}{(x+2)(a+2)},\{\forall x\ne a\}.

\text{This expression } \rightarrow -\frac{3}{(a+2)^{2}} \text{ as } x \rightarrow a.

Example 5

\frac{\frac{x+2}{x}-\frac{a+2}{a}}{x-a} = \frac{\frac{x+2}{x}\left(\frac{a}{a}\right)-\left(\frac{x}{x}\right)\frac{a+2}{a}}{x-a} = \frac{\frac{ax+2a-(ax+2x)}{ax}}{x-a} = \frac{2(a-x)}{(x-a)ax} = -\frac{2}{ax}, \{\forall x \ne a\}.

\text{This expression } \rightarrow -\frac{2}{a^2} \text{ as } x \rightarrow a.

Example 6

\frac{4x^{2}(x-3)^{1/2}-x^{3}(2)(x-3)^{-1/2}(4)}{(x-3)^{3/2}}=\frac{4x^{2}(x-3)^{1/2}-2x^{3}(x-3)^{-1/2}}{(x-3)^{3/2}}=\frac{4x^{2}(x-3)-2x^{3}}{(x-3)^{3/2}}

=\frac{2x^{3}-12x^{2}}{(x-3)^{3/2}}=\frac{2x^{2}(x-6)}{(x-3)^{3/2}},\{\forall x>3\}.

\text{This expression is 0 when } x=6.

This expression is 0 when x = 6.

Example 7

\frac{(3x+2)^{1/2}(2x+3)^{-1}(2)-(2x+3)(\frac{1}{2})(3x+2)^{-1/2}}{(3x+2)^{1/2}} = \frac{\frac{2(3x+2)^{1/2}}{2x+3} - \frac{2x+3}{2(3x+2)^{1/2}}}{(3x+2)^{1/2}}

= \frac{\frac{4(3x+2)-(2x+3)^{2}}{2(2x+3)(3x+2)^{1/2}}}{(3x+2)^{1/2}} = -\frac{12x+8-4x^2-12x-9}{2(2x+3)(3x+2)} = -\frac{4x^{2}+1}{2(2x+3)(3x+2)}, \left\{\forall x > -\frac{2}{3}\right\}.

There are no values of x for which this expression is equal to 0.

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