Find the Equation Given the Roots 1/2 , 1/4 , 1/5

$\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{5}$

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 $x = \frac{1}{2}$ was found by solving for $x$ when $x - (\frac{1}{2}) = y$ and $y = 0$ .

The factor is $x - \frac{1}{2}$

Step 3

The root at $x = \frac{1}{4}$ was found by solving for $x$ when $x - (\frac{1}{4}) = y$ and $y = 0$ .

The factor is $x - \frac{1}{4}$

Step 4

The root at $x = \frac{1}{5}$ was found by solving for $x$ when $x - (\frac{1}{5}) = y$ and $y = 0$ .

The factor is $x - \frac{1}{5}$

Step 5

Combine all the factors into a single equation.

$y = (x - \frac{1}{2}) (x - \frac{1}{4}) (x - \frac{1}{5})$

Step 6

Multiply all the factors to simplify the equation $y = (x - \frac{1}{2}) (x - \frac{1}{4}) (x - \frac{1}{5})$ .

Expand $(x - \frac{1}{2}) (x - \frac{1}{4})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Combine $x$ and $\frac{1}{4}$ .

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

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

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

Multiply $- \frac{1}{2} (- \frac{1}{4})$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{1}{2}$ by $1$ .

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

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

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

Multiply $2$ by $4$ .

$y = (x^{2} - \frac{x}{4} - \frac{x}{2} + \frac{1}{8}) (x - \frac{1}{5})$

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

$y = (x^{2} - \frac{x}{4} - \frac{x}{2} \cdot \frac{2}{2} + \frac{1}{8}) (x - \frac{1}{5})$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{2}$ by $\frac{2}{2}$ .

$y = (x^{2} - \frac{x}{4} - \frac{x \cdot 2}{2 \cdot 2} + \frac{1}{8}) (x - \frac{1}{5})$

Multiply $2$ by $2$ .

$y = (x^{2} - \frac{x}{4} - \frac{x \cdot 2}{4} + \frac{1}{8}) (x - \frac{1}{5})$

Combine the numerators over the common denominator.

$y = (x^{2} + \frac{- x - x \cdot 2}{4} + \frac{1}{8}) (x - \frac{1}{5})$

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

$y = (x^{2} \cdot \frac{4}{4} + \frac{- x - x \cdot 2}{4} + \frac{1}{8}) (x - \frac{1}{5})$

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

$y = (\frac{x^{2} \cdot 4}{4} + \frac{- x - x \cdot 2}{4} + \frac{1}{8}) (x - \frac{1}{5})$

Combine the numerators over the common denominator.

$y = (\frac{x^{2} \cdot 4 - x - x \cdot 2}{4} + \frac{1}{8}) (x - \frac{1}{5})$

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

$y = (\frac{x^{2} \cdot 4 - x - x \cdot 2}{4} \cdot \frac{2}{2} + \frac{1}{8}) (x - \frac{1}{5})$

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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

$y = (\frac{(x^{2} \cdot 4 - x - x \cdot 2) \cdot 2}{4 \cdot 2} + \frac{1}{8}) (x - \frac{1}{5})$

Multiply $4$ by $2$ .

$y = (\frac{(x^{2} \cdot 4 - x - x \cdot 2) \cdot 2}{8} + \frac{1}{8}) (x - \frac{1}{5})$

Combine the numerators over the common denominator.

$y = \frac{(x^{2} \cdot 4 - x - x \cdot 2) \cdot 2 + 1}{8} \cdot (x - \frac{1}{5})$

Simplify the numerator.

Simplify each term.

Move $4$ to the left of $x^{2}$ .

$y = \frac{(4 \cdot x^{2} - x - x \cdot 2) \cdot 2 + 1}{8} \cdot (x - \frac{1}{5})$

Multiply $2$ by $- 1$ .

$y = \frac{(4 x^{2} - x - 2 x) \cdot 2 + 1}{8} \cdot (x - \frac{1}{5})$

Subtract $2 x$ from $- x$ .

$y = \frac{(4 x^{2} - 3 x) \cdot 2 + 1}{8} \cdot (x - \frac{1}{5})$

Apply the distributive property.

$y = \frac{4 x^{2} \cdot 2 - 3 x \cdot 2 + 1}{8} \cdot (x - \frac{1}{5})$

Multiply $2$ by $4$ .

$y = \frac{8 x^{2} - 3 x \cdot 2 + 1}{8} \cdot (x - \frac{1}{5})$

Multiply $2$ by $- 3$ .

$y = \frac{8 x^{2} - 6 x + 1}{8} \cdot (x - \frac{1}{5})$

Factor by grouping.

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 $a \cdot c = 8 \cdot 1 = 8$ and whose sum is $b = - 6$ .

Factor $- 6$ out of $- 6 x$ .

$y = \frac{8 x^{2} - 6 x + 1}{8} \cdot (x - \frac{1}{5})$

Rewrite $- 6$ as $- 2$ plus $- 4$

$y = \frac{8 x^{2} + (- 2 - 4) x + 1}{8} \cdot (x - \frac{1}{5})$

Apply the distributive property.

$y = \frac{8 x^{2} - 2 x - 4 x + 1}{8} \cdot (x - \frac{1}{5})$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y = \frac{(8 x^{2} - 2 x) - 4 x + 1}{8} \cdot (x - \frac{1}{5})$

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

$y = \frac{2 x (4 x - 1) - (4 x - 1)}{8} \cdot (x - \frac{1}{5})$

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$y = \frac{(4 x - 1) (2 x - 1)}{8} \cdot (x - \frac{1}{5})$

Simplify terms.

Apply the distributive property.

$y = \frac{(4 x - 1) (2 x - 1)}{8} \cdot x + \frac{(4 x - 1) (2 x - 1)}{8} \cdot (- \frac{1}{5})$

Combine $\frac{(4 x - 1) (2 x - 1)}{8}$ and $x$ .

$y = \frac{(4 x - 1) (2 x - 1) x}{8} + \frac{(4 x - 1) (2 x - 1)}{8} \cdot (- \frac{1}{5})$

Multiply $\frac{(4 x - 1) (2 x - 1)}{8} (- \frac{1}{5})$ .

Multiply $\frac{(4 x - 1) (2 x - 1)}{8}$ by $\frac{1}{5}$ .

$y = \frac{(4 x - 1) (2 x - 1) x}{8} - \frac{(4 x - 1) (2 x - 1)}{8 \cdot 5}$

Multiply $8$ by $5$ .

$y = \frac{(4 x - 1) (2 x - 1) x}{8} - \frac{(4 x - 1) (2 x - 1)}{40}$

To write $\frac{(4 x - 1) (2 x - 1) x}{8}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{(4 x - 1) (2 x - 1) x}{8} \cdot \frac{5}{5} - \frac{(4 x - 1) (2 x - 1)}{40}$

Write each expression with a common denominator of $40$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{(4 x - 1) (2 x - 1) x}{8}$ by $\frac{5}{5}$ .

$y = \frac{(4 x - 1) (2 x - 1) x \cdot 5}{8 \cdot 5} - \frac{(4 x - 1) (2 x - 1)}{40}$

Multiply $8$ by $5$ .

$y = \frac{(4 x - 1) (2 x - 1) x \cdot 5}{40} - \frac{(4 x - 1) (2 x - 1)}{40}$

Combine the numerators over the common denominator.

$y = \frac{(4 x - 1) (2 x - 1) x \cdot 5 - (4 x - 1) (2 x - 1)}{40}$

Simplify the numerator.

Factor $(4 x - 1) (2 x - 1)$ out of $(4 x - 1) (2 x - 1) x \cdot 5 - (4 x - 1) (2 x - 1)$ .

Factor $(4 x - 1) (2 x - 1)$ out of $(4 x - 1) (2 x - 1) x \cdot 5$ .

$y = \frac{(4 x - 1) (2 x - 1) (x \cdot 5) - (4 x - 1) (2 x - 1)}{40}$

Factor $(4 x - 1) (2 x - 1)$ out of $- (4 x - 1) (2 x - 1)$ .

$y = \frac{(4 x - 1) (2 x - 1) (x \cdot 5) + (4 x - 1) (2 x - 1) (- 1)}{40}$

Factor $(4 x - 1) (2 x - 1)$ out of $(4 x - 1) (2 x - 1) (x \cdot 5) + (4 x - 1) (2 x - 1) (- 1)$ .

$y = \frac{(4 x - 1) (2 x - 1) (x \cdot 5 - 1)}{40}$

Move $5$ to the left of $x$ .

$y = \frac{(4 x - 1) (2 x - 1) (5 x - 1)}{40}$

Expand $(4 x - 1) (2 x - 1)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{(4 x (2 x - 1) - 1 (2 x - 1)) (5 x - 1)}{40}$

Apply the distributive property.

$y = \frac{(4 x (2 x) + 4 x \cdot - 1 - 1 (2 x - 1)) (5 x - 1)}{40}$

Apply the distributive property.

$y = \frac{(4 x (2 x) + 4 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (5 x - 1)}{40}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$y = \frac{(4 \cdot (2 x \cdot x) + 4 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (5 x - 1)}{40}$

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

Move $x$ .

$y = \frac{(4 \cdot (2 (x \cdot x)) + 4 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (5 x - 1)}{40}$

Multiply $x$ by $x$ .

$y = \frac{(4 \cdot (2 x^{2}) + 4 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (5 x - 1)}{40}$

Multiply $4$ by $2$ .

$y = \frac{(8 x^{2} + 4 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (5 x - 1)}{40}$

Multiply $- 1$ by $4$ .

$y = \frac{(8 x^{2} - 4 x - 1 (2 x) - 1 \cdot - 1) (5 x - 1)}{40}$

Multiply $2$ by $- 1$ .

$y = \frac{(8 x^{2} - 4 x - 2 x - 1 \cdot - 1) (5 x - 1)}{40}$

Multiply $- 1$ by $- 1$ .

$y = \frac{(8 x^{2} - 4 x - 2 x + 1) (5 x - 1)}{40}$

Subtract $2 x$ from $- 4 x$ .

$y = \frac{(8 x^{2} - 6 x + 1) (5 x - 1)}{40}$

Expand $(8 x^{2} - 6 x + 1) (5 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$y = \frac{8 x^{2} (5 x) + 8 x^{2} \cdot - 1 - 6 x (5 x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

Simplify each term.

Rewrite using the commutative property of multiplication.

$y = \frac{8 \cdot (5 x^{2} x) + 8 x^{2} \cdot - 1 - 6 x (5 x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

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

Move $x$ .

$y = \frac{8 \cdot (5 (x \cdot x^{2})) + 8 x^{2} \cdot - 1 - 6 x (5 x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

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

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

$y = \frac{8 \cdot (5 (x \cdot x^{2})) + 8 x^{2} \cdot - 1 - 6 x (5 x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

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

$y = \frac{8 \cdot (5 x^{1 + 2}) + 8 x^{2} \cdot - 1 - 6 x (5 x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

Add $1$ and $2$ .

$y = \frac{8 \cdot (5 x^{3}) + 8 x^{2} \cdot - 1 - 6 x (5 x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

Multiply $8$ by $5$ .

$y = \frac{40 x^{3} + 8 x^{2} \cdot - 1 - 6 x (5 x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

Multiply $- 1$ by $8$ .

$y = \frac{40 x^{3} - 8 x^{2} - 6 x (5 x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

Rewrite using the commutative property of multiplication.

$y = \frac{40 x^{3} - 8 x^{2} - 6 \cdot (5 x \cdot x) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

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

Move $x$ .

$y = \frac{40 x^{3} - 8 x^{2} - 6 \cdot (5 (x \cdot x)) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

Multiply $x$ by $x$ .

$y = \frac{40 x^{3} - 8 x^{2} - 6 \cdot (5 x^{2}) - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

Multiply $- 6$ by $5$ .

$y = \frac{40 x^{3} - 8 x^{2} - 30 x^{2} - 6 x \cdot - 1 + 1 (5 x) + 1 \cdot - 1}{40}$

Multiply $- 1$ by $- 6$ .

$y = \frac{40 x^{3} - 8 x^{2} - 30 x^{2} + 6 x + 1 (5 x) + 1 \cdot - 1}{40}$

Multiply $5 x$ by $1$ .

$y = \frac{40 x^{3} - 8 x^{2} - 30 x^{2} + 6 x + 5 x + 1 \cdot - 1}{40}$

Multiply $- 1$ by $1$ .

$y = \frac{40 x^{3} - 8 x^{2} - 30 x^{2} + 6 x + 5 x - 1}{40}$

Subtract $30 x^{2}$ from $- 8 x^{2}$ .

$y = \frac{40 x^{3} - 38 x^{2} + 6 x + 5 x - 1}{40}$

Add $6 x$ and $5 x$ .

$y = \frac{40 x^{3} - 38 x^{2} + 11 x - 1}{40}$

Split the fraction $\frac{40 x^{3} - 38 x^{2} + 11 x - 1}{40}$ into two fractions.

$y = \frac{40 x^{3} - 38 x^{2} + 11 x}{40} + \frac{- 1}{40}$

Split the fraction $\frac{40 x^{3} - 38 x^{2} + 11 x}{40}$ into two fractions.

$y = \frac{40 x^{3} - 38 x^{2}}{40} + \frac{11 x}{40} + \frac{- 1}{40}$

Split the fraction $\frac{40 x^{3} - 38 x^{2}}{40}$ into two fractions.

$y = \frac{40 x^{3}}{40} + \frac{- 38 x^{2}}{40} + \frac{11 x}{40} + \frac{- 1}{40}$

Cancel the common factor of $40$ .

Cancel the common factor.

$y = \frac{40 x^{3}}{40} + \frac{- 38 x^{2}}{40} + \frac{11 x}{40} + \frac{- 1}{40}$

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

$y = x^{3} + \frac{- 38 x^{2}}{40} + \frac{11 x}{40} + \frac{- 1}{40}$

Cancel the common factor of $- 38$ and $40$ .

Factor $2$ out of $- 38 x^{2}$ .

$y = x^{3} + \frac{2 (- 19 x^{2})}{40} + \frac{11 x}{40} + \frac{- 1}{40}$

Cancel the common factors.

Factor $2$ out of $40$ .

$y = x^{3} + \frac{2 (- 19 x^{2})}{2 (20)} + \frac{11 x}{40} + \frac{- 1}{40}$

Cancel the common factor.

$y = x^{3} + \frac{2 (- 19 x^{2})}{2 \cdot 20} + \frac{11 x}{40} + \frac{- 1}{40}$

Rewrite the expression.

$y = x^{3} + \frac{- 19 x^{2}}{20} + \frac{11 x}{40} + \frac{- 1}{40}$

Move the negative in front of the fraction.

$y = x^{3} - \frac{19 x^{2}}{20} + \frac{11 x}{40} + \frac{- 1}{40}$

Move the negative in front of the fraction.

$y = x^{3} - \frac{19 x^{2}}{20} + \frac{11 x}{40} - \frac{1}{40}$

Step 7

Find the Equation Given the Roots 1/2 , 1/4 , 1/5

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