Algebra Examples

$f (y) = \frac{2 y^{3} - y^{2} + 2 y - 1}{y^{3} - y^{2} + y - 1}$ , $g (y) = \frac{2 y^{2} - 3 y + 1}{4 y^{2} - 4 y + 1}$

Step 1

Replace the function designators with the actual functions in $f (y) \cdot (g (y))$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Factor out the greatest common factor from each group.

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Step 2.1.1.1

Group the first two terms and the last two terms.

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

Step 2.1.1.2

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

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

Step 2.1.2

Factor the polynomial by factoring out the greatest common factor, $2 y - 1$ .

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

Factor out the greatest common factor from each group.

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Step 2.2.1.1

Group the first two terms and the last two terms.

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

Step 2.2.1.2

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

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

Step 2.2.2

Factor the polynomial by factoring out the greatest common factor, $y - 1$ .

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

Step 2.3

Factor by grouping.

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Step 2.3.1

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 $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = - 3$ .

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Step 2.3.1.1

Factor $- 3$ out of $- 3 y$ .

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

Step 2.3.1.2

Rewrite $- 3$ as $- 1$ plus $- 2$

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Factor out the greatest common factor from each group.

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Step 2.3.2.1

Group the first two terms and the last two terms.

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

Step 2.3.2.2

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

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

Step 2.3.3

Factor the polynomial by factoring out the greatest common factor, $2 y - 1$ .

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

Step 2.4

Factor using the perfect square rule.

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Step 2.4.1

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

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

Step 2.4.2

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

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

Step 2.4.3

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

$4 y = 2 \cdot (2 y) \cdot 1$

Step 2.4.4

Rewrite the polynomial.

$\frac{(2 y - 1) (y^{2} + 1)}{(y - 1) (y^{2} + 1)} \cdot \frac{(2 y - 1) (y - 1)}{{(2 y)}^{2} - 2 \cdot (2 y) \cdot 1 + 1^{2}}$

Step 2.4.5

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

$\frac{(2 y - 1) (y^{2} + 1)}{(y - 1) (y^{2} + 1)} \cdot \frac{(2 y - 1) (y - 1)}{{(2 y - 1)}^{2}}$

Step 2.5

Simplify terms.

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Step 2.5.1

Combine.

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

Step 2.5.2

Cancel the common factor of $2 y - 1$ and ${(2 y - 1)}^{2}$ .

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Step 2.5.2.1

Factor $2 y - 1$ out of $(2 y - 1) (y^{2} + 1) ((2 y - 1) (y - 1))$ .

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

Step 2.5.2.2

Cancel the common factors.

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Step 2.5.2.2.1

Factor $2 y - 1$ out of $(y - 1) (y^{2} + 1) {(2 y - 1)}^{2}$ .

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

Step 2.5.2.2.2

Cancel the common factor.

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

Step 2.5.2.2.3

Rewrite the expression.

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

Step 2.5.3

Cancel the common factor of $y^{2} + 1$ .

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Step 2.5.3.1

Cancel the common factor.

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

Step 2.5.3.2

Rewrite the expression.

$\frac{(2 y - 1) (y - 1)}{(y - 1) (2 y - 1)}$

Step 2.5.4

Cancel the common factor of $2 y - 1$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{(2 y - 1) (y - 1)}{(y - 1) (2 y - 1)}$

Step 2.5.4.2

Rewrite the expression.

$\frac{y - 1}{y - 1}$

Step 2.5.5

Cancel the common factor of $y - 1$ .

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Step 2.5.5.1

Cancel the common factor.

$\frac{y - 1}{y - 1}$

Step 2.5.5.2

Rewrite the expression.

$1$