Find the Equation Given the Roots 7 , 6-i

$7$ , $6 - i$

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 $x = 7$ was found by solving for $x$ when $x - (7) = y$ and $y = 0$ .

The factor is $x - 7$

Step 3

The root at $x = 6 - i$ was found by solving for $x$ when $x - (6 - i) = y$ and $y = 0$ .

The factor is $x - 6 + i$

Step 4

The root at $x = 6 + i$ was found by solving for $x$ when $x - (6 + i) = y$ and $y = 0$ .

The factor is $x - 6 - i$

Step 5

Combine all the factors into a single equation.

$y = (x - 7) (x - 6 + i) (x - 6 - i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 7) (x - 6 + i) (x - 6 - i)$ .

Expand $(x - 7) (x - 6 + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 6 + x i - 7 x - 7 \cdot - 6 - 7 i) (x - 6 - i)$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 6$ to the left of $x$ .

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

Multiply $- 7$ by $- 6$ .

$y = (x^{2} - 6 x + x i - 7 x + 42 - 7 i) (x - 6 - i)$

Subtract $7 x$ from $- 6 x$ .

$y = (x^{2} + x i - 13 x + 42 - 7 i) (x - 6 - i)$

Expand $(x^{2} + x i - 13 x + 42 - 7 i) (x - 6 - i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 6 + x^{2} (- i) + x i x + x i \cdot - 6 + x i (- i) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Simplify terms.

Simplify each term.

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

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

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

$y = x^{2} x + x^{2} \cdot - 6 + x^{2} (- i) + x i x + x i \cdot - 6 + x i (- i) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

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

$y = x^{2 + 1} + x^{2} \cdot - 6 + x^{2} (- i) + x i x + x i \cdot - 6 + x i (- i) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 6 + x^{2} (- i) + x i x + x i \cdot - 6 + x i (- i) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Move $- 6$ to the left of $x^{2}$ .

$y = x^{3} - 6 \cdot x^{2} + x^{2} (- i) + x i x + x i \cdot - 6 + x i (- i) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

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

Move $x$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x \cdot x i + x i \cdot - 6 + x i (- i) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $x$ by $x$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i + x i \cdot - 6 + x i (- i) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Move $- 6$ to the left of $x i$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 \cdot (x i) + x i (- i) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $x i (- i)$ .

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

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x (- (i i)) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

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

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x (- (i i)) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

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

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x (- i^{1 + 1}) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Add $1$ and $1$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x (- i^{2}) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

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

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x (1) - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $- 1$ by $- 1$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x \cdot 1 - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $x$ by $1$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x \cdot x - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

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

Move $x$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 (x \cdot x) - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $x$ by $x$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} - 13 x \cdot - 6 - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $- 6$ by $- 13$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x - 13 x (- i) + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $- 1$ by $- 13$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x + 42 \cdot - 6 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $42$ by $- 6$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 + 42 (- i) - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $- 1$ by $42$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x - 7 i \cdot - 6 - 7 i (- i)$

Multiply $- 6$ by $- 7$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i - 7 i (- i)$

Multiply $- 7 i (- i)$ .

Multiply $- 1$ by $- 7$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i + 7 i i$

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

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i + 7 (i i)$

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

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i + 7 (i i)$

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

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i + 7 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i + 7 i^{2}$

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

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i + 7 \cdot - 1$

Multiply $7$ by $- 1$ .

$y = x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i - 7$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 6 x^{2} + x^{2} (- i) + x^{2} i - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i - 7$ .

Reorder the factors in the terms $x^{2} (- i)$ and $x^{2} i$ .

$y = x^{3} - 6 x^{2} - i x^{2} + i x^{2} - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i - 7$

Add $- i x^{2}$ and $i x^{2}$ .

$y = x^{3} - 6 x^{2} + 0 - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i - 7$

Add $x^{3} - 6 x^{2}$ and $0$ .

$y = x^{3} - 6 x^{2} - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 42 i - 7 i x + 42 i - 7$

Add $- 42 i$ and $42 i$ .

$y = x^{3} - 6 x^{2} - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 7 i x + 0 - 7$

Add $x^{3} - 6 x^{2} - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 7 i x$ and $0$ .

$y = x^{3} - 6 x^{2} - 6 x i + x - 13 x^{2} + 78 x + 13 x i + 42 x - 252 - 7 i x - 7$

Subtract $13 x^{2}$ from $- 6 x^{2}$ .

$y = x^{3} - 19 x^{2} - 6 x i + x + 78 x + 13 x i + 42 x - 252 - 7 i x - 7$

Add $- 6 x i$ and $13 x i$ .

$y = x^{3} - 19 x^{2} + 7 x i + x + 78 x + 42 x - 252 - 7 i x - 7$

Combine the opposite terms in $x^{3} - 19 x^{2} + 7 x i + x + 78 x + 42 x - 252 - 7 i x - 7$ .

Reorder the factors in the terms $7 x i$ and $- 7 i x$ .

$y = x^{3} - 19 x^{2} + 7 i x + x + 78 x + 42 x - 252 - 7 i x - 7$

Subtract $7 i x$ from $7 i x$ .

$y = x^{3} - 19 x^{2} + x + 78 x + 42 x - 252 + 0 - 7$

Add $x^{3} - 19 x^{2} + x + 78 x + 42 x - 252$ and $0$ .

$y = x^{3} - 19 x^{2} + x + 78 x + 42 x - 252 - 7$

Add $x$ and $78 x$ .

$y = x^{3} - 19 x^{2} + 79 x + 42 x - 252 - 7$

Add $79 x$ and $42 x$ .

$y = x^{3} - 19 x^{2} + 121 x - 252 - 7$

Subtract $7$ from $- 252$ .

$y = x^{3} - 19 x^{2} + 121 x - 259$

Step 7

Find the Equation Given the Roots 7 , 6-i

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