Simplify (4r)/(r^2-1)-(3r+1)/(r^2-2r+1)

$\frac{4 r}{r^{2} - 1} - \frac{3 r + 1}{r^{2} - 2 r + 1}$

Step 1

Simplify each term.

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$\frac{4 r}{r^{2} - 1^{2}} - \frac{3 r + 1}{r^{2} - 2 r + 1}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r$ and $b = 1$ .

$\frac{4 r}{(r + 1) (r - 1)} - \frac{3 r + 1}{r^{2} - 2 r + 1}$

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$\frac{4 r}{(r + 1) (r - 1)} - \frac{3 r + 1}{r^{2} - 2 r + 1^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 r = 2 \cdot r \cdot 1$

Rewrite the polynomial.

$\frac{4 r}{(r + 1) (r - 1)} - \frac{3 r + 1}{r^{2} - 2 \cdot r \cdot 1 + 1^{2}}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = r$ and $b = 1$ .

$\frac{4 r}{(r + 1) (r - 1)} - \frac{3 r + 1}{{(r - 1)}^{2}}$

Step 2

To write $\frac{4 r}{(r + 1) (r - 1)}$ as a fraction with a common denominator, multiply by $\frac{r - 1}{r - 1}$ .

$\frac{4 r}{(r + 1) (r - 1)} \cdot \frac{r - 1}{r - 1} - \frac{3 r + 1}{{(r - 1)}^{2}}$

Step 3

To write $- \frac{3 r + 1}{{(r - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{r + 1}{r + 1}$ .

$\frac{4 r}{(r + 1) (r - 1)} \cdot \frac{r - 1}{r - 1} - \frac{3 r + 1}{{(r - 1)}^{2}} \cdot \frac{r + 1}{r + 1}$

Step 4

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

Multiply $\frac{4 r}{(r + 1) (r - 1)}$ by $\frac{r - 1}{r - 1}$ .

$\frac{4 r (r - 1)}{(r + 1) (r - 1) (r - 1)} - \frac{3 r + 1}{{(r - 1)}^{2}} \cdot \frac{r + 1}{r + 1}$

Raise $r - 1$ to the power of $1$ .

$\frac{4 r (r - 1)}{(r + 1) ({(r - 1)}^{1} (r - 1))} - \frac{3 r + 1}{{(r - 1)}^{2}} \cdot \frac{r + 1}{r + 1}$

Raise $r - 1$ to the power of $1$ .

$\frac{4 r (r - 1)}{(r + 1) ({(r - 1)}^{1} {(r - 1)}^{1})} - \frac{3 r + 1}{{(r - 1)}^{2}} \cdot \frac{r + 1}{r + 1}$

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

$\frac{4 r (r - 1)}{(r + 1) {(r - 1)}^{1 + 1}} - \frac{3 r + 1}{{(r - 1)}^{2}} \cdot \frac{r + 1}{r + 1}$

Add $1$ and $1$ .

$\frac{4 r (r - 1)}{(r + 1) {(r - 1)}^{2}} - \frac{3 r + 1}{{(r - 1)}^{2}} \cdot \frac{r + 1}{r + 1}$

Multiply $\frac{3 r + 1}{{(r - 1)}^{2}}$ by $\frac{r + 1}{r + 1}$ .

$\frac{4 r (r - 1)}{(r + 1) {(r - 1)}^{2}} - \frac{(3 r + 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Reorder the factors of $(r + 1) {(r - 1)}^{2}$ .

$\frac{4 r (r - 1)}{{(r - 1)}^{2} (r + 1)} - \frac{(3 r + 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 r (r - 1) - (3 r + 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{4 r \cdot r + 4 r \cdot - 1 - (3 r + 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Multiply $r$ by $r$ by adding the exponents.

Move $r$ .

$\frac{4 (r \cdot r) + 4 r \cdot - 1 - (3 r + 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Multiply $r$ by $r$ .

$\frac{4 r^{2} + 4 r \cdot - 1 - (3 r + 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Multiply $- 1$ by $4$ .

$\frac{4 r^{2} - 4 r - (3 r + 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Apply the distributive property.

$\frac{4 r^{2} - 4 r + (- (3 r) - 1 \cdot 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Multiply $3$ by $- 1$ .

$\frac{4 r^{2} - 4 r + (- 3 r - 1 \cdot 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Multiply $- 1$ by $1$ .

$\frac{4 r^{2} - 4 r + (- 3 r - 1) (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Expand $(- 3 r - 1) (r + 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{4 r^{2} - 4 r - 3 r (r + 1) - 1 (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Apply the distributive property.

$\frac{4 r^{2} - 4 r - 3 r \cdot r - 3 r \cdot 1 - 1 (r + 1)}{{(r - 1)}^{2} (r + 1)}$

Apply the distributive property.

$\frac{4 r^{2} - 4 r - 3 r \cdot r - 3 r \cdot 1 - 1 r - 1 \cdot 1}{{(r - 1)}^{2} (r + 1)}$

Simplify and combine like terms.

Simplify each term.

Multiply $r$ by $r$ by adding the exponents.

Move $r$ .

$\frac{4 r^{2} - 4 r - 3 (r \cdot r) - 3 r \cdot 1 - 1 r - 1 \cdot 1}{{(r - 1)}^{2} (r + 1)}$

Multiply $r$ by $r$ .

$\frac{4 r^{2} - 4 r - 3 r^{2} - 3 r \cdot 1 - 1 r - 1 \cdot 1}{{(r - 1)}^{2} (r + 1)}$

Multiply $- 3$ by $1$ .

$\frac{4 r^{2} - 4 r - 3 r^{2} - 3 r - 1 r - 1 \cdot 1}{{(r - 1)}^{2} (r + 1)}$

Rewrite $- 1 r$ as $- r$ .

$\frac{4 r^{2} - 4 r - 3 r^{2} - 3 r - r - 1 \cdot 1}{{(r - 1)}^{2} (r + 1)}$

Multiply $- 1$ by $1$ .

$\frac{4 r^{2} - 4 r - 3 r^{2} - 3 r - r - 1}{{(r - 1)}^{2} (r + 1)}$

Subtract $r$ from $- 3 r$ .

$\frac{4 r^{2} - 4 r - 3 r^{2} - 4 r - 1}{{(r - 1)}^{2} (r + 1)}$

Subtract $3 r^{2}$ from $4 r^{2}$ .

$\frac{r^{2} - 4 r - 4 r - 1}{{(r - 1)}^{2} (r + 1)}$

Subtract $4 r$ from $- 4 r$ .

$\frac{r^{2} - 8 r - 1}{{(r - 1)}^{2} (r + 1)}$

Simplify (4r)/(r^2-1)-(3r+1)/(r^2-2r+1)

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