Algebra Examples

$- 64$ , $0.001$ , $\frac{27}{125}$

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 $x = - 64$ was found by solving for $x$ when $x - (- 64) = y$ and $y = 0$ .

The factor is $x + 64$

Step 3

The root at $x = 0.001$ was found by solving for $x$ when $x - (0.001) = y$ and $y = 0$ .

The factor is $x - 0.001$

Step 4

The root at $x = \frac{27}{125}$ was found by solving for $x$ when $x - (\frac{27}{125}) = y$ and $y = 0$ .

The factor is $x - \frac{27}{125}$

Step 5

Combine all the factors into a single equation.

$y = (x + 64) (x - 0.001) (x - \frac{27}{125})$

Step 6

Multiply all the factors to simplify the equation $y = (x + 64) (x - 0.001) (x - \frac{27}{125})$ .

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Step 6.1

Expand $(x + 64) (x - 0.001)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$y = (x (x - 0.001) + 64 (x - 0.001)) (x - \frac{27}{125})$

Step 6.1.2

Apply the distributive property.

$y = (x \cdot x + x \cdot - 0.001 + 64 (x - 0.001)) (x - \frac{27}{125})$

Step 6.1.3

Apply the distributive property.

$y = (x \cdot x + x \cdot - 0.001 + 64 x + 64 \cdot - 0.001) (x - \frac{27}{125})$

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 $x$ .

$y = (x^{2} + x \cdot - 0.001 + 64 x + 64 \cdot - 0.001) (x - \frac{27}{125})$

Step 6.2.1.2

Move $- 0.001$ to the left of $x$ .

$y = (x^{2} - 0.001 \cdot x + 64 x + 64 \cdot - 0.001) (x - \frac{27}{125})$

Step 6.2.1.3

Multiply $64$ by $- 0.001$ .

$y = (x^{2} - 0.001 x + 64 x - 0.064) (x - \frac{27}{125})$

Step 6.2.2

Add $- 0.001 x$ and $64 x$ .

$y = (x^{2} + 63.999 x - 0.064) (x - \frac{27}{125})$

Step 6.3

Expand $(x^{2} + 63.999 x - 0.064) (x - \frac{27}{125})$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} (- \frac{27}{125}) + 63.999 x \cdot x + 63.999 x (- \frac{27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4

Simplify each term.

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Step 6.4.1

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

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Step 6.4.1.1

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

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Step 6.4.1.1.1

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

$y = x^{2} x + x^{2} (- \frac{27}{125}) + 63.999 x \cdot x + 63.999 x (- \frac{27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.1.1.2

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

$y = x^{2 + 1} + x^{2} (- \frac{27}{125}) + 63.999 x \cdot x + 63.999 x (- \frac{27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.1.2

Add $2$ and $1$ .

$y = x^{3} + x^{2} (- \frac{27}{125}) + 63.999 x \cdot x + 63.999 x (- \frac{27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.2

Combine $x^{2}$ and $\frac{27}{125}$ .

$y = x^{3} - \frac{x^{2} \cdot 27}{125} + 63.999 x \cdot x + 63.999 x (- \frac{27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.3

Move $27$ to the left of $x^{2}$ .

$y = x^{3} - \frac{27 \cdot x^{2}}{125} + 63.999 x \cdot x + 63.999 x (- \frac{27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.4

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

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Step 6.4.4.1

Move $x$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 (x \cdot x) + 63.999 x (- \frac{27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.4.2

Multiply $x$ by $x$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} + 63.999 x (- \frac{27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.5

Multiply $63.999 x (- \frac{27}{125})$ .

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Step 6.4.5.1

Multiply $- 1$ by $63.999$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - 63.999 x \frac{27}{125} - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.5.2

Combine $- 63.999$ and $\frac{27}{125}$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} + x (\frac{- 63.999 \cdot 27}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.5.3

Multiply $- 63.999$ by $27$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} + x (\frac{- 1727.973}{125}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.5.4

Combine $x$ and $\frac{- 1727.973}{125}$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} + \frac{x \cdot - 1727.973}{125} - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.6

Move $- 1727.973$ to the left of $x$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} + \frac{- 1727.973 \cdot x}{125} - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.7

Move the negative in front of the fraction.

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - \frac{(1727.973) \cdot x}{125} - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.8

Factor $1727.973$ out of $1727.973 \cdot x$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - \frac{1727.973 (x)}{125} - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.9

Factor $125$ out of $125$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - \frac{1727.973 (x)}{125 (1)} - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.10

Separate fractions.

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - (\frac{1727.973}{125} \cdot \frac{x}{1}) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.11

Divide $1727.973$ by $125$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - (13.823784 (\frac{x}{1})) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.12

Divide $x$ by $1$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - (13.823784 x) - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.13

Multiply $13.823784$ by $- 1$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - 13.823784 x - 0.064 x - 0.064 (- \frac{27}{125})$

Step 6.4.14

Multiply $- 0.064 (- \frac{27}{125})$ .

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Step 6.4.14.1

Multiply $- 1$ by $- 0.064$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - 13.823784 x - 0.064 x + 0.064 (\frac{27}{125})$

Step 6.4.14.2

Combine $0.064$ and $\frac{27}{125}$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - 13.823784 x - 0.064 x + \frac{0.064 \cdot 27}{125}$

Step 6.4.14.3

Multiply $0.064$ by $27$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - 13.823784 x - 0.064 x + \frac{1.728}{125}$

Step 6.4.15

Divide $1.728$ by $125$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} - 13.823784 x - 0.064 x + 0.013824$

Step 6.5

To write $63.999 x^{2}$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

$y = x^{3} - \frac{27 x^{2}}{125} + 63.999 x^{2} \cdot \frac{125}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.6

Simplify terms.

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Step 6.6.1

Combine $63.999 x^{2}$ and $\frac{125}{125}$ .

$y = x^{3} - \frac{27 x^{2}}{125} + \frac{63.999 x^{2} \cdot 125}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.6.2

Combine the numerators over the common denominator.

$y = x^{3} + \frac{- 27 x^{2} + 63.999 x^{2} \cdot 125}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7

Simplify each term.

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Step 6.7.1

Simplify the numerator.

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Step 6.7.1.1

Factor $0.009 x^{2}$ out of $- 27 x^{2} + 63.999 x^{2} \cdot 125$ .

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Step 6.7.1.1.1

Move $x^{2}$ .

$y = x^{3} + \frac{- 27 x^{2} + 63.999 \cdot (125 x^{2})}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.1.1.2

Factor $0.009 x^{2}$ out of $- 27 x^{2}$ .

$y = x^{3} + \frac{0.009 x^{2} (- 3000) + 63.999 \cdot (125 x^{2})}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.1.1.3

Factor $0.009 x^{2}$ out of $63.999 \cdot 125 x^{2}$ .

$y = x^{3} + \frac{0.009 x^{2} (- 3000) + 0.009 x^{2} (7111 \cdot 125)}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.1.1.4

Factor $0.009 x^{2}$ out of $0.009 x^{2} (- 3000) + 0.009 x^{2} (7111 \cdot 125)$ .

$y = x^{3} + \frac{0.009 x^{2} (- 3000 + 7111 \cdot 125)}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.1.2

Combine exponents.

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Step 6.7.1.2.1

Multiply $7111$ by $125$ .

$y = x^{3} + \frac{0.009 x^{2} (- 3000 + 888875)}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.1.2.2

Multiply $- 3000 + 888875$ by $0.009$ .

$y = x^{3} + \frac{7972.875 x^{2}}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.2

Factor $7972.875$ out of $7972.875 x^{2}$ .

$y = x^{3} + \frac{7972.875 (x^{2})}{125} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.3

Factor $125$ out of $125$ .

$y = x^{3} + \frac{7972.875 (x^{2})}{125 (1)} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.4

Separate fractions.

$y = x^{3} + \frac{7972.875}{125} \cdot \frac{x^{2}}{1} - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.5

Divide $7972.875$ by $125$ .

$y = x^{3} + 63.783 (\frac{x^{2}}{1}) - 13.823784 x - 0.064 x + 0.013824$

Step 6.7.6

Divide $x^{2}$ by $1$ .

$y = x^{3} + 63.783 x^{2} - 13.823784 x - 0.064 x + 0.013824$

Step 6.8

Subtract $0.064 x$ from $- 13.823784 x$ .

$y = x^{3} + 63.783 x^{2} - 13.887784 x + 0.013824$

Step 7