Algebra Examples

$- \frac{7}{4}$ , $\sqrt{5}$ , $- \sqrt{5}$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = - \frac{7}{4}$ was found by solving for $x$ when $x - (- \frac{7}{4}) = y$ and $y = 0$ .

The factor is $x + \frac{7}{4}$

Step 3

The root at $x = \sqrt{5}$ was found by solving for $x$ when $x - (\sqrt{5}) = y$ and $y = 0$ .

The factor is $x - \sqrt{5}$

Step 4

The root at $x = - \sqrt{5}$ was found by solving for $x$ when $x - (- \sqrt{5}) = y$ and $y = 0$ .

The factor is $x + \sqrt{5}$

Step 5

Combine all the factors into a single equation.

$y = (x + \frac{7}{4}) (x - \sqrt{5}) (x + \sqrt{5})$

Step 6

Multiply all the factors to simplify the equation $y = (x + \frac{7}{4}) (x - \sqrt{5}) (x + \sqrt{5})$ .

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Step 6.1

Expand $(x + \frac{7}{4}) (x - \sqrt{5})$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$y = (x (x - \sqrt{5}) + \frac{7}{4} \cdot (x - \sqrt{5})) (x + \sqrt{5})$

Step 6.1.2

Apply the distributive property.

$y = (x \cdot x + x (- \sqrt{5}) + \frac{7}{4} \cdot (x - \sqrt{5})) (x + \sqrt{5})$

Step 6.1.3

Apply the distributive property.

$y = (x \cdot x + x (- \sqrt{5}) + \frac{7}{4} x + \frac{7}{4} \cdot (- \sqrt{5})) (x + \sqrt{5})$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

$y = (x^{2} + x (- \sqrt{5}) + \frac{7}{4} x + \frac{7}{4} \cdot (- \sqrt{5})) (x + \sqrt{5})$

Step 6.2.1.2

Combine $\frac{7}{4}$ and $x$ .

$y = (x^{2} + x (- \sqrt{5}) + \frac{7 x}{4} + \frac{7}{4} \cdot (- \sqrt{5})) (x + \sqrt{5})$

Step 6.2.1.3

Combine $\frac{7}{4}$ and $\sqrt{5}$ .

$y = (x^{2} + x (- \sqrt{5}) + \frac{7 x}{4} - \frac{7 \sqrt{5}}{4}) (x + \sqrt{5})$

Step 6.2.2

Subtract $\frac{7 \sqrt{5}}{4}$ from $x (- \sqrt{5})$ .

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Step 6.2.2.1

Reorder $x$ and $- 1$ .

$y = (x^{2} + \frac{7 x}{4} - 1 \cdot (x \sqrt{5}) - \frac{7 \sqrt{5}}{4}) (x + \sqrt{5})$

Step 6.2.2.2

To write $- 1 \cdot x \sqrt{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = (x^{2} + \frac{7 x}{4} - 1 \cdot (x \sqrt{5}) \cdot \frac{4}{4} - \frac{7 \sqrt{5}}{4}) (x + \sqrt{5})$

Step 6.2.2.3

Combine $- 1 \cdot x \sqrt{5}$ and $\frac{4}{4}$ .

$y = (x^{2} + \frac{7 x}{4} + \frac{- 1 \cdot (x \sqrt{5}) \cdot 4}{4} - \frac{7 \sqrt{5}}{4}) (x + \sqrt{5})$

Step 6.2.2.4

Combine the numerators over the common denominator.

$y = (x^{2} + \frac{7 x}{4} + \frac{- 1 \cdot (x \sqrt{5}) \cdot 4 - 7 \sqrt{5}}{4}) (x + \sqrt{5})$

Step 6.2.3

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

$y = (\frac{7 x}{4} + x^{2} \cdot \frac{4}{4} + \frac{- 1 \cdot (x \sqrt{5}) \cdot 4 - 7 \sqrt{5}}{4}) (x + \sqrt{5})$

Step 6.2.4

Combine $x^{2}$ and $\frac{4}{4}$ .

$y = (\frac{7 x}{4} + \frac{x^{2} \cdot 4}{4} + \frac{- 1 \cdot (x \sqrt{5}) \cdot 4 - 7 \sqrt{5}}{4}) (x + \sqrt{5})$

Step 6.2.5

Combine the numerators over the common denominator.

$y = (\frac{7 x}{4} + \frac{x^{2} \cdot 4 - 1 \cdot (x \sqrt{5}) \cdot 4 - 7 \sqrt{5}}{4}) (x + \sqrt{5})$

Step 6.2.6

Combine the numerators over the common denominator.

$y = \frac{7 x + x^{2} \cdot 4 - 1 \cdot (x \sqrt{5}) \cdot 4 - 7 \sqrt{5}}{4} \cdot (x + \sqrt{5})$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Reorder terms.

$y = \frac{4 x^{2} - 1 \cdot (4 x \sqrt{5}) + 7 x - 7 \sqrt{5}}{4} \cdot (x + \sqrt{5})$

Step 6.3.2

Factor out the greatest common factor from each group.

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Step 6.3.2.1

Group the first two terms and the last two terms.

$y = \frac{(4 x^{2} - 1 \cdot (4 x \sqrt{5})) + 7 x - 7 \sqrt{5}}{4} \cdot (x + \sqrt{5})$

Step 6.3.2.2

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

$y = \frac{4 x (x - \sqrt{5}) + 7 (x - \sqrt{5})}{4} \cdot (x + \sqrt{5})$

Step 6.3.3

Factor the polynomial by factoring out the greatest common factor, $x - \sqrt{5}$ .

$y = \frac{(x - \sqrt{5}) (4 x + 7)}{4} \cdot (x + \sqrt{5})$

Step 6.4

Simplify terms.

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Step 6.4.1

Apply the distributive property.

$y = \frac{(x - \sqrt{5}) (4 x + 7)}{4} \cdot x + \frac{(x - \sqrt{5}) (4 x + 7)}{4} \cdot \sqrt{5}$

Step 6.4.2

Combine fractions.

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Step 6.4.2.1

Combine $\frac{(x - \sqrt{5}) (4 x + 7)}{4}$ and $x$ .

$y = \frac{(x - \sqrt{5}) (4 x + 7) x}{4} + \frac{(x - \sqrt{5}) (4 x + 7)}{4} \cdot \sqrt{5}$

Step 6.4.2.2

Combine $\frac{(x - \sqrt{5}) (4 x + 7)}{4}$ and $\sqrt{5}$ .

$y = \frac{(x - \sqrt{5}) (4 x + 7) x}{4} + \frac{(x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.2.3

Combine the numerators over the common denominator.

$y = \frac{(x - \sqrt{5}) (4 x + 7) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3

Simplify each term.

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Step 6.4.3.1

Expand $(x - \sqrt{5}) (4 x + 7)$ using the FOIL Method.

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Step 6.4.3.1.1

Apply the distributive property.

$y = \frac{(x (4 x + 7) - \sqrt{5} (4 x + 7)) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.1.2

Apply the distributive property.

$y = \frac{(x (4 x) + x \cdot 7 - \sqrt{5} (4 x + 7)) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.1.3

Apply the distributive property.

$y = \frac{(x (4 x) + x \cdot 7 - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.2

Simplify each term.

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Step 6.4.3.2.1

Rewrite using the commutative property of multiplication.

$y = \frac{(4 x \cdot x + x \cdot 7 - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.2.2

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

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Step 6.4.3.2.2.1

Move $x$ .

$y = \frac{(4 (x \cdot x) + x \cdot 7 - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.2.2.2

Multiply $x$ by $x$ .

$y = \frac{(4 x^{2} + x \cdot 7 - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.2.3

Move $7$ to the left of $x$ .

$y = \frac{(4 x^{2} + 7 \cdot x - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.2.4

Multiply $4$ by $- 1$ .

$y = \frac{(4 x^{2} + 7 x - 4 \sqrt{5} x - \sqrt{5} \cdot 7) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.2.5

Multiply $7$ by $- 1$ .

$y = \frac{(4 x^{2} + 7 x - 4 \sqrt{5} x - 7 \sqrt{5}) x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.3

Apply the distributive property.

$y = \frac{4 x^{2} x + 7 x \cdot x - 4 \sqrt{5} x \cdot x - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.4

Simplify.

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Step 6.4.3.4.1

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

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Step 6.4.3.4.1.1

Move $x$ .

$y = \frac{4 (x \cdot x^{2}) + 7 x \cdot x - 4 \sqrt{5} x \cdot x - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.4.1.2

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

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Step 6.4.3.4.1.2.1

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

$y = \frac{4 (x \cdot x^{2}) + 7 x \cdot x - 4 \sqrt{5} x \cdot x - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.4.1.2.2

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

$y = \frac{4 x^{1 + 2} + 7 x \cdot x - 4 \sqrt{5} x \cdot x - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.4.1.3

Add $1$ and $2$ .

$y = \frac{4 x^{3} + 7 x \cdot x - 4 \sqrt{5} x \cdot x - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.4.2

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

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Step 6.4.3.4.2.1

Move $x$ .

$y = \frac{4 x^{3} + 7 (x \cdot x) - 4 \sqrt{5} x \cdot x - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.4.2.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x \cdot x - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.4.3

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

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Step 6.4.3.4.3.1

Move $x$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} (x \cdot x) - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.4.3.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (x - \sqrt{5}) (4 x + 7) \sqrt{5}}{4}$

Step 6.4.3.5

Expand $(x - \sqrt{5}) (4 x + 7)$ using the FOIL Method.

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Step 6.4.3.5.1

Apply the distributive property.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (x (4 x + 7) - \sqrt{5} (4 x + 7)) \sqrt{5}}{4}$

Step 6.4.3.5.2

Apply the distributive property.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (x (4 x) + x \cdot 7 - \sqrt{5} (4 x + 7)) \sqrt{5}}{4}$

Step 6.4.3.5.3

Apply the distributive property.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (x (4 x) + x \cdot 7 - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) \sqrt{5}}{4}$

Step 6.4.3.6

Simplify each term.

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Step 6.4.3.6.1

Rewrite using the commutative property of multiplication.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (4 x \cdot x + x \cdot 7 - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) \sqrt{5}}{4}$

Step 6.4.3.6.2

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

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Step 6.4.3.6.2.1

Move $x$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (4 (x \cdot x) + x \cdot 7 - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) \sqrt{5}}{4}$

Step 6.4.3.6.2.2

Multiply $x$ by $x$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (4 x^{2} + x \cdot 7 - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) \sqrt{5}}{4}$

Step 6.4.3.6.3

Move $7$ to the left of $x$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (4 x^{2} + 7 \cdot x - \sqrt{5} (4 x) - \sqrt{5} \cdot 7) \sqrt{5}}{4}$

Step 6.4.3.6.4

Multiply $4$ by $- 1$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (4 x^{2} + 7 x - 4 \sqrt{5} x - \sqrt{5} \cdot 7) \sqrt{5}}{4}$

Step 6.4.3.6.5

Multiply $7$ by $- 1$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + (4 x^{2} + 7 x - 4 \sqrt{5} x - 7 \sqrt{5}) \sqrt{5}}{4}$

Step 6.4.3.7

Apply the distributive property.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 \sqrt{5} x \sqrt{5} - 7 \sqrt{5} \sqrt{5}}{4}$

Step 6.4.3.8

Simplify.

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Step 6.4.3.8.1

Multiply $- 4 \sqrt{5} x \sqrt{5}$ .

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Step 6.4.3.8.1.1

Raise $\sqrt{5}$ to the power of $1$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x (\sqrt{5} \sqrt{5}) - 7 \sqrt{5} \sqrt{5}}{4}$

Step 6.4.3.8.1.2

Raise $\sqrt{5}$ to the power of $1$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x (\sqrt{5} \sqrt{5}) - 7 \sqrt{5} \sqrt{5}}{4}$

Step 6.4.3.8.1.3

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

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x {\sqrt{5}}^{1 + 1} - 7 \sqrt{5} \sqrt{5}}{4}$

Step 6.4.3.8.1.4

Add $1$ and $1$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x {\sqrt{5}}^{2} - 7 \sqrt{5} \sqrt{5}}{4}$

Step 6.4.3.8.2

Multiply $- 7 \sqrt{5} \sqrt{5}$ .

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Step 6.4.3.8.2.1

Raise $\sqrt{5}$ to the power of $1$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x {\sqrt{5}}^{2} - 7 (\sqrt{5} \sqrt{5})}{4}$

Step 6.4.3.8.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x {\sqrt{5}}^{2} - 7 (\sqrt{5} \sqrt{5})}{4}$

Step 6.4.3.8.2.3

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

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x {\sqrt{5}}^{2} - 7 {\sqrt{5}}^{1 + 1}}{4}$

Step 6.4.3.8.2.4

Add $1$ and $1$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x {\sqrt{5}}^{2} - 7 {\sqrt{5}}^{2}}{4}$

Step 6.4.3.9

Simplify each term.

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Step 6.4.3.9.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.4.3.9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x {(5^{\frac{1}{2}})}^{2} - 7 {\sqrt{5}}^{2}}{4}$

Step 6.4.3.9.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x \cdot 5^{\frac{1}{2} \cdot 2} - 7 {\sqrt{5}}^{2}}{4}$

Step 6.4.3.9.1.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x \cdot 5^{\frac{2}{2}} - 7 {\sqrt{5}}^{2}}{4}$

Step 6.4.3.9.1.4

Cancel the common factor of $2$ .

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Step 6.4.3.9.1.4.1

Cancel the common factor.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x \cdot 5^{\frac{2}{2}} - 7 {\sqrt{5}}^{2}}{4}$

Step 6.4.3.9.1.4.2

Rewrite the expression.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x \cdot 5 - 7 {\sqrt{5}}^{2}}{4}$

Step 6.4.3.9.1.5

Evaluate the exponent.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 4 x \cdot 5 - 7 {\sqrt{5}}^{2}}{4}$

Step 6.4.3.9.2

Multiply $5$ by $- 4$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 7 {\sqrt{5}}^{2}}{4}$

Step 6.4.3.9.3

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.4.3.9.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 7 {(5^{\frac{1}{2}})}^{2}}{4}$

Step 6.4.3.9.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 7 \cdot 5^{\frac{1}{2} \cdot 2}}{4}$

Step 6.4.3.9.3.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 7 \cdot 5^{\frac{2}{2}}}{4}$

Step 6.4.3.9.3.4

Cancel the common factor of $2$ .

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Step 6.4.3.9.3.4.1

Cancel the common factor.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 7 \cdot 5^{\frac{2}{2}}}{4}$

Step 6.4.3.9.3.4.2

Rewrite the expression.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 7 \cdot 5}{4}$

Step 6.4.3.9.3.5

Evaluate the exponent.

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 7 \cdot 5}{4}$

Step 6.4.3.9.4

Multiply $- 7$ by $5$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 35}{4}$

Step 6.4.4

Combine the opposite terms in $4 x^{3} + 7 x^{2} - 4 \sqrt{5} x^{2} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 35$ .

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Step 6.4.4.1

Reorder the factors in the terms $- 4 \sqrt{5} x^{2}$ and $4 x^{2} \sqrt{5}$ .

$y = \frac{4 x^{3} + 7 x^{2} - 4 x^{2} \sqrt{5} - 7 \sqrt{5} x + 4 x^{2} \sqrt{5} + 7 x \sqrt{5} - 20 x - 35}{4}$

Step 6.4.4.2

Add $- 4 x^{2} \sqrt{5}$ and $4 x^{2} \sqrt{5}$ .

$y = \frac{4 x^{3} + 7 x^{2} - 7 \sqrt{5} x + 0 + 7 x \sqrt{5} - 20 x - 35}{4}$

Step 6.4.4.3

Add $4 x^{3} + 7 x^{2} - 7 \sqrt{5} x$ and $0$ .

$y = \frac{4 x^{3} + 7 x^{2} - 7 \sqrt{5} x + 7 x \sqrt{5} - 20 x - 35}{4}$

Step 6.4.4.4

Reorder the factors in the terms $- 7 \sqrt{5} x$ and $7 x \sqrt{5}$ .

$y = \frac{4 x^{3} + 7 x^{2} - 7 x \sqrt{5} + 7 x \sqrt{5} - 20 x - 35}{4}$

Step 6.4.4.5

Add $- 7 x \sqrt{5}$ and $7 x \sqrt{5}$ .

$y = \frac{4 x^{3} + 7 x^{2} + 0 - 20 x - 35}{4}$

Step 6.4.4.6

Add $4 x^{3} + 7 x^{2}$ and $0$ .

$y = \frac{4 x^{3} + 7 x^{2} - 20 x - 35}{4}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Factor out the greatest common factor from each group.

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Step 6.5.1.1

Group the first two terms and the last two terms.

$y = \frac{(4 x^{3} + 7 x^{2}) - 20 x - 35}{4}$

Step 6.5.1.2

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

$y = \frac{x^{2} (4 x + 7) - 5 (4 x + 7)}{4}$

Step 6.5.2

Factor the polynomial by factoring out the greatest common factor, $4 x + 7$ .

$y = \frac{(4 x + 7) (x^{2} - 5)}{4}$

Step 6.6

Expand $(4 x + 7) (x^{2} - 5)$ using the FOIL Method.

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Step 6.6.1

Apply the distributive property.

$y = \frac{4 x (x^{2} - 5) + 7 (x^{2} - 5)}{4}$

Step 6.6.2

Apply the distributive property.

$y = \frac{4 x \cdot x^{2} + 4 x \cdot - 5 + 7 (x^{2} - 5)}{4}$

Step 6.6.3

Apply the distributive property.

$y = \frac{4 x \cdot x^{2} + 4 x \cdot - 5 + 7 x^{2} + 7 \cdot - 5}{4}$

Step 6.7

Simplify each term.

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Step 6.7.1

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

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Step 6.7.1.1

Move $x^{2}$ .

$y = \frac{4 (x^{2} x) + 4 x \cdot - 5 + 7 x^{2} + 7 \cdot - 5}{4}$

Step 6.7.1.2

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

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Step 6.7.1.2.1

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

$y = \frac{4 (x^{2} x) + 4 x \cdot - 5 + 7 x^{2} + 7 \cdot - 5}{4}$

Step 6.7.1.2.2

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

$y = \frac{4 x^{2 + 1} + 4 x \cdot - 5 + 7 x^{2} + 7 \cdot - 5}{4}$

Step 6.7.1.3

Add $2$ and $1$ .

$y = \frac{4 x^{3} + 4 x \cdot - 5 + 7 x^{2} + 7 \cdot - 5}{4}$

Step 6.7.2

Multiply $- 5$ by $4$ .

$y = \frac{4 x^{3} - 20 x + 7 x^{2} + 7 \cdot - 5}{4}$

Step 6.7.3

Multiply $7$ by $- 5$ .

$y = \frac{4 x^{3} - 20 x + 7 x^{2} - 35}{4}$

Step 6.8

Split the fraction $\frac{4 x^{3} - 20 x + 7 x^{2} - 35}{4}$ into two fractions.

$y = \frac{4 x^{3} - 20 x + 7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.9

Split the fraction $\frac{4 x^{3} - 20 x + 7 x^{2}}{4}$ into two fractions.

$y = \frac{4 x^{3} - 20 x}{4} + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.10

Split the fraction $\frac{4 x^{3} - 20 x}{4}$ into two fractions.

$y = \frac{4 x^{3}}{4} + \frac{- 20 x}{4} + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.11

Cancel the common factor of $4$ .

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Step 6.11.1

Cancel the common factor.

$y = \frac{4 x^{3}}{4} + \frac{- 20 x}{4} + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.11.2

Divide $x^{3}$ by $1$ .

$y = x^{3} + \frac{- 20 x}{4} + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.12

Cancel the common factor of $- 20$ and $4$ .

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Step 6.12.1

Factor $4$ out of $- 20 x$ .

$y = x^{3} + \frac{4 (- 5 x)}{4} + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.12.2

Cancel the common factors.

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Step 6.12.2.1

Factor $4$ out of $4$ .

$y = x^{3} + \frac{4 (- 5 x)}{4 (1)} + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.12.2.2

Cancel the common factor.

$y = x^{3} + \frac{4 (- 5 x)}{4 \cdot 1} + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.12.2.3

Rewrite the expression.

$y = x^{3} + \frac{- 5 x}{1} + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.12.2.4

Divide $- 5 x$ by $1$ .

$y = x^{3} - 5 x + \frac{7 x^{2}}{4} + \frac{- 35}{4}$

Step 6.13

Move the negative in front of the fraction.

$y = x^{3} - 5 x + \frac{7 x^{2}}{4} - \frac{35}{4}$

Step 7