Precalculus Examples

Given that $f (x) = x^{2} - 19 x + 90$ and $g (x) = x - 10$ ; find $(f \div g) (x)$ and express the result in standard form.

Step 1

Evaluate $f \div g$ .

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Step 1.1

Replace the function designators in $f \div g$ with the actual functions.

$(x^{2} - 19 x + 90) \div (x - 10)$

Step 1.2

Rewrite the division as a fraction.

$\frac{x^{2} - 19 x + 90}{x - 10}$

Step 1.3

Factor $x^{2} - 19 x + 90$ using the AC method.

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Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $90$ and whose sum is $- 19$ .

$- 10, - 9$

Step 1.3.2

Write the factored form using these integers.

$\frac{(x - 10) (x - 9)}{x - 10}$

Step 1.4

Cancel the common factor of $x - 10$ .

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Step 1.4.1

Cancel the common factor.

$\frac{(x - 10) (x - 9)}{x - 10}$

Step 1.4.2

Divide $x - 9$ by $1$ .

$x - 9$

Step 2

Evaluate $f \div g$ at $x$ .

$(x) - 9$

Step 3

Simplify $(x) - 9$ .

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Step 3.1

Remove parentheses.

$(x) - 9$

Step 3.2

Remove parentheses.

$x - 9$