Find the Equation Given the Roots 3+i , 1

$3 + i$ , $1$

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 $x = 3 + i$ was found by solving for $x$ when $x - (3 + i) = y$ and $y = 0$ .

The factor is $x - 3 - i$

Step 3

The root at $x = 3 - i$ was found by solving for $x$ when $x - (3 - i) = y$ and $y = 0$ .

The factor is $x - 3 + i$

Step 4

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

The factor is $x - 1$

Step 5

Combine all the factors into a single equation.

$y = (x - 3 - i) (x - 3 + i) (x - 1)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 3 - i) (x - 3 + i) (x - 1)$ .

Expand $(x - 3 - i) (x - 3 + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 3 + x i - 3 x - 3 \cdot - 3 - 3 i - i x - i \cdot - 3 - i i) (x - 1)$

Simplify terms.

Combine the opposite terms in $x \cdot x + x \cdot - 3 + x i - 3 x - 3 \cdot - 3 - 3 i - i x - i \cdot - 3 - i i$ .

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

$y = (x \cdot x + x \cdot - 3 + i x - 3 x - 3 \cdot - 3 - 3 i - i x - i \cdot - 3 - i i) (x - 1)$

Subtract $i x$ from $i x$ .

$y = (x \cdot x + x \cdot - 3 + 0 - 3 x - 3 \cdot - 3 - 3 i - i \cdot - 3 - i i) (x - 1)$

Add $x \cdot x + x \cdot - 3$ and $0$ .

$y = (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 - 3 i - i \cdot - 3 - i i) (x - 1)$

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $- 3$ by $- 3$ .

$y = (x^{2} - 3 x - 3 x + 9 - 3 i - i \cdot - 3 - i i) (x - 1)$

Multiply $- 3$ by $- 1$ .

$y = (x^{2} - 3 x - 3 x + 9 - 3 i + 3 i - i i) (x - 1)$

Multiply $- i i$ .

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

$y = (x^{2} - 3 x - 3 x + 9 - 3 i + 3 i - (i i)) (x - 1)$

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

$y = (x^{2} - 3 x - 3 x + 9 - 3 i + 3 i - (i i)) (x - 1)$

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

$y = (x^{2} - 3 x - 3 x + 9 - 3 i + 3 i - i^{1 + 1}) (x - 1)$

Add $1$ and $1$ .

$y = (x^{2} - 3 x - 3 x + 9 - 3 i + 3 i - i^{2}) (x - 1)$

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

$y = (x^{2} - 3 x - 3 x + 9 - 3 i + 3 i + 1) (x - 1)$

Multiply $- 1$ by $- 1$ .

$y = (x^{2} - 3 x - 3 x + 9 - 3 i + 3 i + 1) (x - 1)$

Simplify by adding terms.

Combine the opposite terms in $x^{2} - 3 x - 3 x + 9 - 3 i + 3 i + 1$ .

Add $- 3 i$ and $3 i$ .

$y = (x^{2} - 3 x - 3 x + 9 + 0 + 1) (x - 1)$

Add $x^{2} - 3 x - 3 x + 9$ and $0$ .

$y = (x^{2} - 3 x - 3 x + 9 + 1) (x - 1)$

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

$y = (x^{2} - 6 x + 9 + 1) (x - 1)$

Add $9$ and $1$ .

$y = (x^{2} - 6 x + 10) (x - 1)$

Expand $(x^{2} - 6 x + 10) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1 + 10 x + 10 \cdot - 1$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1 + 10 x + 10 \cdot - 1$

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

$y = x^{2 + 1} + x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1 + 10 x + 10 \cdot - 1$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 1 - 6 x \cdot x - 6 x \cdot - 1 + 10 x + 10 \cdot - 1$

Move $- 1$ to the left of $x^{2}$ .

$y = x^{3} - 1 \cdot x^{2} - 6 x \cdot x - 6 x \cdot - 1 + 10 x + 10 \cdot - 1$

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

$y = x^{3} - x^{2} - 6 x \cdot x - 6 x \cdot - 1 + 10 x + 10 \cdot - 1$

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

Move $x$ .

$y = x^{3} - x^{2} - 6 (x \cdot x) - 6 x \cdot - 1 + 10 x + 10 \cdot - 1$

Multiply $x$ by $x$ .

$y = x^{3} - x^{2} - 6 x^{2} - 6 x \cdot - 1 + 10 x + 10 \cdot - 1$

Multiply $- 1$ by $- 6$ .

$y = x^{3} - x^{2} - 6 x^{2} + 6 x + 10 x + 10 \cdot - 1$

Multiply $10$ by $- 1$ .

$y = x^{3} - x^{2} - 6 x^{2} + 6 x + 10 x - 10$

Simplify by adding terms.

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

$y = x^{3} - 7 x^{2} + 6 x + 10 x - 10$

Add $6 x$ and $10 x$ .

$y = x^{3} - 7 x^{2} + 16 x - 10$

Step 7

Find the Equation Given the Roots 3+i , 1

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