Finite Math Examples

$\frac{4 x - 3}{x^{2} - 9} - \frac{2 x - 3}{x - 3}$

Step 1

To write $\frac{4 x - 3}{x^{2} - 9}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{4 x - 3}{x^{2} - 9} \cdot \frac{x - 3}{x - 3} - \frac{2 x - 3}{x - 3}$

Step 2

To write $- \frac{2 x - 3}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x^{2} - 9}{x^{2} - 9}$ .

$\frac{4 x - 3}{x^{2} - 9} \cdot \frac{x - 3}{x - 3} - \frac{2 x - 3}{x - 3} \cdot \frac{x^{2} - 9}{x^{2} - 9}$

Step 3

Write each expression with a common denominator of $(x^{2} - 9) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{4 x - 3}{x^{2} - 9}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(4 x - 3) (x - 3)}{(x^{2} - 9) (x - 3)} - \frac{2 x - 3}{x - 3} \cdot \frac{x^{2} - 9}{x^{2} - 9}$

Step 3.2

Multiply $\frac{2 x - 3}{x - 3}$ by $\frac{x^{2} - 9}{x^{2} - 9}$ .

$\frac{(4 x - 3) (x - 3)}{(x^{2} - 9) (x - 3)} - \frac{(2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 3.3

Reorder the factors of $(x^{2} - 9) (x - 3)$ .

$\frac{(4 x - 3) (x - 3)}{(x - 3) (x^{2} - 9)} - \frac{(2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(4 x - 3) (x - 3) - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5

Rewrite $\frac{(4 x - 3) (x - 3) - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$ in a factored form.

Tap for more steps...

Step 5.1

Expand $(4 x - 3) (x - 3)$ using the FOIL Method.

Tap for more steps...

Step 5.1.1

Apply the distributive property.

$\frac{4 x (x - 3) - 3 (x - 3) - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.1.2

Apply the distributive property.

$\frac{4 x \cdot x + 4 x \cdot - 3 - 3 (x - 3) - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.1.3

Apply the distributive property.

$\frac{4 x \cdot x + 4 x \cdot - 3 - 3 x - 3 \cdot - 3 - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.2

Simplify and combine like terms.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 5.2.1.1.1

Move $x$ .

$\frac{4 (x \cdot x) + 4 x \cdot - 3 - 3 x - 3 \cdot - 3 - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.2.1.1.2

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 4 x \cdot - 3 - 3 x - 3 \cdot - 3 - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.2.1.2

Multiply $- 3$ by $4$ .

$\frac{4 x^{2} - 12 x - 3 x - 3 \cdot - 3 - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.2.1.3

Multiply $- 3$ by $- 3$ .

$\frac{4 x^{2} - 12 x - 3 x + 9 - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.2.2

Subtract $3 x$ from $- 12 x$ .

$\frac{4 x^{2} - 15 x + 9 - (2 x - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.3

Apply the distributive property.

$\frac{4 x^{2} - 15 x + 9 + (- (2 x) - - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.4

Multiply $2$ by $- 1$ .

$\frac{4 x^{2} - 15 x + 9 + (- 2 x - - 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.5

Multiply $- 1$ by $- 3$ .

$\frac{4 x^{2} - 15 x + 9 + (- 2 x + 3) (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.6

Expand $(- 2 x + 3) (x^{2} - 9)$ using the FOIL Method.

Tap for more steps...

Step 5.6.1

Apply the distributive property.

$\frac{4 x^{2} - 15 x + 9 - 2 x (x^{2} - 9) + 3 (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.6.2

Apply the distributive property.

$\frac{4 x^{2} - 15 x + 9 - 2 x \cdot x^{2} - 2 x \cdot - 9 + 3 (x^{2} - 9)}{(x - 3) (x^{2} - 9)}$

Step 5.6.3

Apply the distributive property.

$\frac{4 x^{2} - 15 x + 9 - 2 x \cdot x^{2} - 2 x \cdot - 9 + 3 x^{2} + 3 \cdot - 9}{(x - 3) (x^{2} - 9)}$

Step 5.7

Simplify each term.

Tap for more steps...

Step 5.7.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 5.7.1.1

Move $x^{2}$ .

$\frac{4 x^{2} - 15 x + 9 - 2 (x^{2} x) - 2 x \cdot - 9 + 3 x^{2} + 3 \cdot - 9}{(x - 3) (x^{2} - 9)}$

Step 5.7.1.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 5.7.1.2.1

Raise $x$ to the power of $1$ .

$\frac{4 x^{2} - 15 x + 9 - 2 (x^{2} x^{1}) - 2 x \cdot - 9 + 3 x^{2} + 3 \cdot - 9}{(x - 3) (x^{2} - 9)}$

Step 5.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 x^{2} - 15 x + 9 - 2 x^{2 + 1} - 2 x \cdot - 9 + 3 x^{2} + 3 \cdot - 9}{(x - 3) (x^{2} - 9)}$

Step 5.7.1.3

Add $2$ and $1$ .

$\frac{4 x^{2} - 15 x + 9 - 2 x^{3} - 2 x \cdot - 9 + 3 x^{2} + 3 \cdot - 9}{(x - 3) (x^{2} - 9)}$

Step 5.7.2

Multiply $- 9$ by $- 2$ .

$\frac{4 x^{2} - 15 x + 9 - 2 x^{3} + 18 x + 3 x^{2} + 3 \cdot - 9}{(x - 3) (x^{2} - 9)}$

Step 5.7.3

Multiply $3$ by $- 9$ .

$\frac{4 x^{2} - 15 x + 9 - 2 x^{3} + 18 x + 3 x^{2} - 27}{(x - 3) (x^{2} - 9)}$

Step 5.8

Add $4 x^{2}$ and $3 x^{2}$ .

$\frac{7 x^{2} - 15 x + 9 - 2 x^{3} + 18 x - 27}{(x - 3) (x^{2} - 9)}$

Step 5.9

Add $- 15 x$ and $18 x$ .

$\frac{7 x^{2} + 3 x + 9 - 2 x^{3} - 27}{(x - 3) (x^{2} - 9)}$

Step 5.10

Subtract $27$ from $9$ .

$\frac{7 x^{2} + 3 x - 2 x^{3} - 18}{(x - 3) (x^{2} - 9)}$

Step 5.11

Reorder terms.

$\frac{- 2 x^{3} + 7 x^{2} + 3 x - 18}{(x - 3) (x^{2} - 9)}$

Step 5.12

Rewrite $- 2 x^{3} + 7 x^{2} + 3 x - 18$ in a factored form.

Tap for more steps...

Step 5.12.1

Factor $- 2 x^{3} + 7 x^{2} + 3 x - 18$ using the rational roots test.

Tap for more steps...

Step 5.12.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$

$q = \pm 1, \pm 2$

Step 5.12.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.5, \pm 18, \pm 9, \pm 2, \pm 4.5, \pm 3, \pm 1.5, \pm 6$

Step 5.12.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Tap for more steps...

Step 5.12.1.3.1

Substitute $2$ into the polynomial.

$- 2 \cdot 2^{3} + 7 \cdot 2^{2} + 3 \cdot 2 - 18$

Step 5.12.1.3.2

Raise $2$ to the power of $3$ .

$- 2 \cdot 8 + 7 \cdot 2^{2} + 3 \cdot 2 - 18$

Step 5.12.1.3.3

Multiply $- 2$ by $8$ .

$- 16 + 7 \cdot 2^{2} + 3 \cdot 2 - 18$

Step 5.12.1.3.4

Raise $2$ to the power of $2$ .

$- 16 + 7 \cdot 4 + 3 \cdot 2 - 18$

Step 5.12.1.3.5

Multiply $7$ by $4$ .

$- 16 + 28 + 3 \cdot 2 - 18$

Step 5.12.1.3.6

Add $- 16$ and $28$ .

$12 + 3 \cdot 2 - 18$

Step 5.12.1.3.7

Multiply $3$ by $2$ .

$12 + 6 - 18$

Step 5.12.1.3.8

Add $12$ and $6$ .

$18 - 18$

Step 5.12.1.3.9

Subtract $18$ from $18$ .

$0$

Step 5.12.1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 2 x^{3} + 7 x^{2} + 3 x - 18}{x - 2}$

Step 5.12.1.5

Divide $- 2 x^{3} + 7 x^{2} + 3 x - 18$ by $x - 2$ .

Tap for more steps...

Step 5.12.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$2 x^{3}$

$7 x^{2}$

$3 x$

$18$

Step 5.12.1.5.2

Divide the highest order term in the dividend $- 2 x^{3}$ by the highest order term in divisor $x$ .

				-	$2 x^{2}$
	$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$

Step 5.12.1.5.3

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			-	$2 x^{3}$	+	$4 x^{2}$

Step 5.12.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} + 4 x^{2}$

			-	$2 x^{2}$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$

Step 5.12.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$2 x^{2}$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$

Step 5.12.1.5.6

Pull the next terms from the original dividend down into the current dividend.

			-	$2 x^{2}$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$

Step 5.12.1.5.7

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

			-	$2 x^{2}$	+	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$

Step 5.12.1.5.8

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$	+	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$
					+	$3 x^{2}$	-	$6 x$

Step 5.12.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} - 6 x$

			-	$2 x^{2}$	+	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$
					-	$3 x^{2}$	+	$6 x$

Step 5.12.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$2 x^{2}$	+	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$
					-	$3 x^{2}$	+	$6 x$
							+	$9 x$

Step 5.12.1.5.11

Pull the next terms from the original dividend down into the current dividend.

			-	$2 x^{2}$	+	$3 x$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$
					-	$3 x^{2}$	+	$6 x$
							+	$9 x$	-	$18$

Step 5.12.1.5.12

Divide the highest order term in the dividend $9 x$ by the highest order term in divisor $x$ .

			-	$2 x^{2}$	+	$3 x$	+	$9$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$
					-	$3 x^{2}$	+	$6 x$
							+	$9 x$	-	$18$

Step 5.12.1.5.13

Multiply the new quotient term by the divisor.

			-	$2 x^{2}$	+	$3 x$	+	$9$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$
					-	$3 x^{2}$	+	$6 x$
							+	$9 x$	-	$18$
							+	$9 x$	-	$18$

Step 5.12.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $9 x - 18$

			-	$2 x^{2}$	+	$3 x$	+	$9$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$
					-	$3 x^{2}$	+	$6 x$
							+	$9 x$	-	$18$
							-	$9 x$	+	$18$

Step 5.12.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$2 x^{2}$	+	$3 x$	+	$9$
$x$	-	$2$	-	$2 x^{3}$	+	$7 x^{2}$	+	$3 x$	-	$18$
			+	$2 x^{3}$	-	$4 x^{2}$
					+	$3 x^{2}$	+	$3 x$
					-	$3 x^{2}$	+	$6 x$
							+	$9 x$	-	$18$
							-	$9 x$	+	$18$
										$0$

Step 5.12.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$- 2 x^{2} + 3 x + 9$

Step 5.12.1.6

Write $- 2 x^{3} + 7 x^{2} + 3 x - 18$ as a set of factors.

$\frac{(x - 2) (- 2 x^{2} + 3 x + 9)}{(x - 3) (x^{2} - 9)}$

Step 5.12.2

Factor by grouping.

Tap for more steps...

Step 5.12.2.1

Factor by grouping.

Tap for more steps...

Step 5.12.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 2 \cdot 9 = - 18$ and whose sum is $b = 3$ .

Tap for more steps...

Step 5.12.2.1.1.1

Factor $3$ out of $3 x$ .

$\frac{(x - 2) (- 2 x^{2} + 3 (x) + 9)}{(x - 3) (x^{2} - 9)}$

Step 5.12.2.1.1.2

Rewrite $3$ as $- 3$ plus $6$

$\frac{(x - 2) (- 2 x^{2} + (- 3 + 6) x + 9)}{(x - 3) (x^{2} - 9)}$

Step 5.12.2.1.1.3

Apply the distributive property.

$\frac{(x - 2) (- 2 x^{2} - 3 x + 6 x + 9)}{(x - 3) (x^{2} - 9)}$

Step 5.12.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.12.2.1.2.1

Group the first two terms and the last two terms.

$\frac{(x - 2) ((- 2 x^{2} - 3 x) + 6 x + 9)}{(x - 3) (x^{2} - 9)}$

Step 5.12.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(x - 2) (x (- 2 x - 3) - 3 (- 2 x - 3))}{(x - 3) (x^{2} - 9)}$

Step 5.12.2.1.3

Factor the polynomial by factoring out the greatest common factor, $- 2 x - 3$ .

$\frac{(x - 2) ((- 2 x - 3) (x - 3))}{(x - 3) (x^{2} - 9)}$

Step 5.12.2.2

Remove unnecessary parentheses.

$\frac{(x - 2) (- 2 x - 3) (x - 3)}{(x - 3) (x^{2} - 9)}$

Step 5.13

Rewrite $9$ as $3^{2}$ .

$\frac{(x - 2) (- 2 x - 3) (x - 3)}{(x - 3) (x^{2} - 3^{2})}$

Step 5.14

Factor.

Tap for more steps...

Step 5.14.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{(x - 2) (- 2 x - 3) (x - 3)}{(x - 3) ((x + 3) (x - 3))}$

Step 5.14.2

Remove unnecessary parentheses.

$\frac{(x - 2) (- 2 x - 3) (x - 3)}{(x - 3) (x + 3) (x - 3)}$

Step 5.15

Combine exponents.

Tap for more steps...

Step 5.15.1

Raise $x - 3$ to the power of $1$ .

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

Step 5.15.2

Raise $x - 3$ to the power of $1$ .

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

Step 5.15.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.15.4

Add $1$ and $1$ .

$\frac{(x - 2) (- 2 x - 3) (x - 3)}{{(x - 3)}^{2} (x + 3)}$

Step 5.16

Reduce the expression $\frac{(x - 2) (- 2 x - 3) (x - 3)}{{(x - 3)}^{2} (x + 3)}$ by cancelling the common factors.

Tap for more steps...

Step 5.16.1

Factor $x - 3$ out of $(x - 2) (- 2 x - 3) (x - 3)$ .

$\frac{(x - 3) ((x - 2) (- 2 x - 3))}{{(x - 3)}^{2} (x + 3)}$

Step 5.16.2

Factor $x - 3$ out of ${(x - 3)}^{2} (x + 3)$ .

$\frac{(x - 3) ((x - 2) (- 2 x - 3))}{(x - 3) ((x - 3) (x + 3))}$

Step 5.16.3

Cancel the common factor.

$\frac{(x - 3) ((x - 2) (- 2 x - 3))}{(x - 3) ((x - 3) (x + 3))}$

Step 5.16.4

Rewrite the expression.

$\frac{(x - 2) (- 2 x - 3)}{(x - 3) (x + 3)}$

Step 6

Factor.

Tap for more steps...

Step 6.1

Factor $- 1$ out of $- 2 x - 3$ .

Tap for more steps...

Step 6.1.1

Factor $- 1$ out of $- 2 x$ .

$\frac{(x - 2) (- (2 x) - 3)}{(x - 3) (x + 3)}$

Step 6.1.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 6.1.3

Factor $- 1$ out of $- (2 x) - 1 (3)$ .

$\frac{(x - 2) (- (2 x + 3))}{(x - 3) (x + 3)}$

Step 6.2

Remove unnecessary parentheses.

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

Step 7

Factor out negative.

$\frac{- ((x - 2) (2 x + 3))}{(x - 3) (x + 3)}$

Step 8

Remove unnecessary parentheses.

$\frac{- (x - 2) (2 x + 3)}{(x - 3) (x + 3)}$