Find the Equation Given the Roots 1/2 , 1/3 , 1/7

$\frac{1}{2}$ , $\frac{1}{3}$ , $\frac{1}{7}$

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 $x = \frac{1}{3}$ was found by solving for $x$ when $x - (\frac{1}{3}) = y$ and $y = 0$ .

The factor is $x - \frac{1}{3}$

Step 4

The root at $x = \frac{1}{7}$ was found by solving for $x$ when $x - (\frac{1}{7}) = y$ and $y = 0$ .

The factor is $x - \frac{1}{7}$

Step 5

Combine all the factors into a single equation.

$y = (x - \frac{1}{2}) (x - \frac{1}{3}) (x - \frac{1}{7})$

Step 6

Multiply all the factors to simplify the equation $y = (x - \frac{1}{2}) (x - \frac{1}{3}) (x - \frac{1}{7})$ .

Expand $(x - \frac{1}{2}) (x - \frac{1}{3})$ using the FOIL Method.

Apply the distributive property.

$y = (x (x - \frac{1}{3}) - \frac{1}{2} \cdot (x - \frac{1}{3})) (x - \frac{1}{7})$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Combine $x$ and $\frac{1}{3}$ .

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

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

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

Multiply $- \frac{1}{2} (- \frac{1}{3})$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

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

Multiply $2$ by $3$ .

$y = (x^{2} - \frac{x}{3} - \frac{x}{2} + \frac{1}{6}) (x - \frac{1}{7})$

To write $- \frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = (x^{2} - \frac{x}{3} \cdot \frac{2}{2} - \frac{x}{2} + \frac{1}{6}) (x - \frac{1}{7})$

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = (x^{2} - \frac{x}{3} \cdot \frac{2}{2} - \frac{x}{2} \cdot \frac{3}{3} + \frac{1}{6}) (x - \frac{1}{7})$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{3}$ by $\frac{2}{2}$ .

$y = (x^{2} - \frac{x \cdot 2}{3 \cdot 2} - \frac{x}{2} \cdot \frac{3}{3} + \frac{1}{6}) (x - \frac{1}{7})$

Multiply $3$ by $2$ .

$y = (x^{2} - \frac{x \cdot 2}{6} - \frac{x}{2} \cdot \frac{3}{3} + \frac{1}{6}) (x - \frac{1}{7})$

Multiply $\frac{x}{2}$ by $\frac{3}{3}$ .

$y = (x^{2} - \frac{x \cdot 2}{6} - \frac{x \cdot 3}{2 \cdot 3} + \frac{1}{6}) (x - \frac{1}{7})$

Multiply $2$ by $3$ .

$y = (x^{2} - \frac{x \cdot 2}{6} - \frac{x \cdot 3}{6} + \frac{1}{6}) (x - \frac{1}{7})$

Combine the numerators over the common denominator.

$y = (x^{2} + \frac{- x \cdot 2 - x \cdot 3}{6} + \frac{1}{6}) (x - \frac{1}{7})$

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = (x^{2} \cdot \frac{6}{6} + \frac{- x \cdot 2 - x \cdot 3}{6} + \frac{1}{6}) (x - \frac{1}{7})$

Combine $x^{2}$ and $\frac{6}{6}$ .

$y = (\frac{x^{2} \cdot 6}{6} + \frac{- x \cdot 2 - x \cdot 3}{6} + \frac{1}{6}) (x - \frac{1}{7})$

Combine the numerators over the common denominator.

$y = (\frac{x^{2} \cdot 6 - x \cdot 2 - x \cdot 3}{6} + \frac{1}{6}) (x - \frac{1}{7})$

Combine the numerators over the common denominator.

$y = \frac{x^{2} \cdot 6 - x \cdot 2 - x \cdot 3 + 1}{6} \cdot (x - \frac{1}{7})$

Simplify the numerator.

Reorder terms.

$y = \frac{6 x^{2} - 1 \cdot (2 x) - 1 \cdot (3 x) + 1}{6} \cdot (x - \frac{1}{7})$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$y = \frac{(6 x^{2} - 1 \cdot (2 x)) - 1 \cdot (3 x) + 1}{6} \cdot (x - \frac{1}{7})$

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

$y = \frac{2 x (3 x - 1) - (3 x - 1)}{6} \cdot (x - \frac{1}{7})$

Factor the polynomial by factoring out the greatest common factor, $3 x - 1$ .

$y = \frac{(3 x - 1) (2 x - 1)}{6} \cdot (x - \frac{1}{7})$

Simplify terms.

Apply the distributive property.

$y = \frac{(3 x - 1) (2 x - 1)}{6} \cdot x + \frac{(3 x - 1) (2 x - 1)}{6} \cdot (- \frac{1}{7})$

Combine $\frac{(3 x - 1) (2 x - 1)}{6}$ and $x$ .

$y = \frac{(3 x - 1) (2 x - 1) x}{6} + \frac{(3 x - 1) (2 x - 1)}{6} \cdot (- \frac{1}{7})$

Multiply $\frac{(3 x - 1) (2 x - 1)}{6} (- \frac{1}{7})$ .

Multiply $\frac{(3 x - 1) (2 x - 1)}{6}$ by $\frac{1}{7}$ .

$y = \frac{(3 x - 1) (2 x - 1) x}{6} - \frac{(3 x - 1) (2 x - 1)}{6 \cdot 7}$

Multiply $6$ by $7$ .

$y = \frac{(3 x - 1) (2 x - 1) x}{6} - \frac{(3 x - 1) (2 x - 1)}{42}$

To write $\frac{(3 x - 1) (2 x - 1) x}{6}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$y = \frac{(3 x - 1) (2 x - 1) x}{6} \cdot \frac{7}{7} - \frac{(3 x - 1) (2 x - 1)}{42}$

Write each expression with a common denominator of $42$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{(3 x - 1) (2 x - 1) x}{6}$ by $\frac{7}{7}$ .

$y = \frac{(3 x - 1) (2 x - 1) x \cdot 7}{6 \cdot 7} - \frac{(3 x - 1) (2 x - 1)}{42}$

Multiply $6$ by $7$ .

$y = \frac{(3 x - 1) (2 x - 1) x \cdot 7}{42} - \frac{(3 x - 1) (2 x - 1)}{42}$

Combine the numerators over the common denominator.

$y = \frac{(3 x - 1) (2 x - 1) x \cdot 7 - (3 x - 1) (2 x - 1)}{42}$

Simplify the numerator.

Factor $(3 x - 1) (2 x - 1)$ out of $(3 x - 1) (2 x - 1) x \cdot 7 - (3 x - 1) (2 x - 1)$ .

Factor $(3 x - 1) (2 x - 1)$ out of $(3 x - 1) (2 x - 1) x \cdot 7$ .

$y = \frac{(3 x - 1) (2 x - 1) (x \cdot 7) - (3 x - 1) (2 x - 1)}{42}$

Factor $(3 x - 1) (2 x - 1)$ out of $- (3 x - 1) (2 x - 1)$ .

$y = \frac{(3 x - 1) (2 x - 1) (x \cdot 7) + (3 x - 1) (2 x - 1) (- 1)}{42}$

Factor $(3 x - 1) (2 x - 1)$ out of $(3 x - 1) (2 x - 1) (x \cdot 7) + (3 x - 1) (2 x - 1) (- 1)$ .

$y = \frac{(3 x - 1) (2 x - 1) (x \cdot 7 - 1)}{42}$

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

$y = \frac{(3 x - 1) (2 x - 1) (7 x - 1)}{42}$

Expand $(3 x - 1) (2 x - 1)$ using the FOIL Method.

Apply the distributive property.

$y = \frac{(3 x (2 x - 1) - 1 (2 x - 1)) (7 x - 1)}{42}$

Apply the distributive property.

$y = \frac{(3 x (2 x) + 3 x \cdot - 1 - 1 (2 x - 1)) (7 x - 1)}{42}$

Apply the distributive property.

$y = \frac{(3 x (2 x) + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (7 x - 1)}{42}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$y = \frac{(3 \cdot (2 x \cdot x) + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (7 x - 1)}{42}$

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

Move $x$ .

$y = \frac{(3 \cdot (2 (x \cdot x)) + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (7 x - 1)}{42}$

Multiply $x$ by $x$ .

$y = \frac{(3 \cdot (2 x^{2}) + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (7 x - 1)}{42}$

Multiply $3$ by $2$ .

$y = \frac{(6 x^{2} + 3 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) (7 x - 1)}{42}$

Multiply $- 1$ by $3$ .

$y = \frac{(6 x^{2} - 3 x - 1 (2 x) - 1 \cdot - 1) (7 x - 1)}{42}$

Multiply $2$ by $- 1$ .

$y = \frac{(6 x^{2} - 3 x - 2 x - 1 \cdot - 1) (7 x - 1)}{42}$

Multiply $- 1$ by $- 1$ .

$y = \frac{(6 x^{2} - 3 x - 2 x + 1) (7 x - 1)}{42}$

Subtract $2 x$ from $- 3 x$ .

$y = \frac{(6 x^{2} - 5 x + 1) (7 x - 1)}{42}$

Expand $(6 x^{2} - 5 x + 1) (7 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$y = \frac{6 x^{2} (7 x) + 6 x^{2} \cdot - 1 - 5 x (7 x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

Simplify each term.

Rewrite using the commutative property of multiplication.

$y = \frac{6 \cdot (7 x^{2} x) + 6 x^{2} \cdot - 1 - 5 x (7 x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

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

Move $x$ .

$y = \frac{6 \cdot (7 (x \cdot x^{2})) + 6 x^{2} \cdot - 1 - 5 x (7 x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

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

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

$y = \frac{6 \cdot (7 (x \cdot x^{2})) + 6 x^{2} \cdot - 1 - 5 x (7 x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

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

$y = \frac{6 \cdot (7 x^{1 + 2}) + 6 x^{2} \cdot - 1 - 5 x (7 x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

Add $1$ and $2$ .

$y = \frac{6 \cdot (7 x^{3}) + 6 x^{2} \cdot - 1 - 5 x (7 x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

Multiply $6$ by $7$ .

$y = \frac{42 x^{3} + 6 x^{2} \cdot - 1 - 5 x (7 x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

Multiply $- 1$ by $6$ .

$y = \frac{42 x^{3} - 6 x^{2} - 5 x (7 x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

Rewrite using the commutative property of multiplication.

$y = \frac{42 x^{3} - 6 x^{2} - 5 \cdot (7 x \cdot x) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

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

Move $x$ .

$y = \frac{42 x^{3} - 6 x^{2} - 5 \cdot (7 (x \cdot x)) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

Multiply $x$ by $x$ .

$y = \frac{42 x^{3} - 6 x^{2} - 5 \cdot (7 x^{2}) - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

Multiply $- 5$ by $7$ .

$y = \frac{42 x^{3} - 6 x^{2} - 35 x^{2} - 5 x \cdot - 1 + 1 (7 x) + 1 \cdot - 1}{42}$

Multiply $- 1$ by $- 5$ .

$y = \frac{42 x^{3} - 6 x^{2} - 35 x^{2} + 5 x + 1 (7 x) + 1 \cdot - 1}{42}$

Multiply $7 x$ by $1$ .

$y = \frac{42 x^{3} - 6 x^{2} - 35 x^{2} + 5 x + 7 x + 1 \cdot - 1}{42}$

Multiply $- 1$ by $1$ .

$y = \frac{42 x^{3} - 6 x^{2} - 35 x^{2} + 5 x + 7 x - 1}{42}$

Subtract $35 x^{2}$ from $- 6 x^{2}$ .

$y = \frac{42 x^{3} - 41 x^{2} + 5 x + 7 x - 1}{42}$

Add $5 x$ and $7 x$ .

$y = \frac{42 x^{3} - 41 x^{2} + 12 x - 1}{42}$

Split the fraction $\frac{42 x^{3} - 41 x^{2} + 12 x - 1}{42}$ into two fractions.

$y = \frac{42 x^{3} - 41 x^{2} + 12 x}{42} + \frac{- 1}{42}$

Split the fraction $\frac{42 x^{3} - 41 x^{2} + 12 x}{42}$ into two fractions.

$y = \frac{42 x^{3} - 41 x^{2}}{42} + \frac{12 x}{42} + \frac{- 1}{42}$

Split the fraction $\frac{42 x^{3} - 41 x^{2}}{42}$ into two fractions.

$y = \frac{42 x^{3}}{42} + \frac{- 41 x^{2}}{42} + \frac{12 x}{42} + \frac{- 1}{42}$

Cancel the common factor of $42$ .

Cancel the common factor.

$y = \frac{42 x^{3}}{42} + \frac{- 41 x^{2}}{42} + \frac{12 x}{42} + \frac{- 1}{42}$

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

$y = x^{3} + \frac{- 41 x^{2}}{42} + \frac{12 x}{42} + \frac{- 1}{42}$

Move the negative in front of the fraction.

$y = x^{3} - \frac{41 x^{2}}{42} + \frac{12 x}{42} + \frac{- 1}{42}$

Cancel the common factor of $12$ and $42$ .

Factor $6$ out of $12 x$ .

$y = x^{3} - \frac{41 x^{2}}{42} + \frac{6 (2 x)}{42} + \frac{- 1}{42}$

Cancel the common factors.

Factor $6$ out of $42$ .

$y = x^{3} - \frac{41 x^{2}}{42} + \frac{6 (2 x)}{6 (7)} + \frac{- 1}{42}$

Cancel the common factor.

$y = x^{3} - \frac{41 x^{2}}{42} + \frac{6 (2 x)}{6 \cdot 7} + \frac{- 1}{42}$

Rewrite the expression.

$y = x^{3} - \frac{41 x^{2}}{42} + \frac{2 x}{7} + \frac{- 1}{42}$

Move the negative in front of the fraction.

$y = x^{3} - \frac{41 x^{2}}{42} + \frac{2 x}{7} - \frac{1}{42}$

Step 7

Find the Equation Given the Roots 1/2 , 1/3 , 1/7

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