Find the Equation Given the Roots 3 , 4+i

$3$ , $4 + i$

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

The factor is $x - 3$

Step 3

The root at $x = 4 + i$ was found by solving for $x$ when $x - (4 + i) = y$ and $y = 0$ .

The factor is $x - 4 - i$

Step 4

The root at $x = 4 - i$ was found by solving for $x$ when $x - (4 - i) = y$ and $y = 0$ .

The factor is $x - 4 + i$

Step 5

Combine all the factors into a single equation.

$y = (x - 3) (x - 4 - i) (x - 4 + i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 3) (x - 4 - i) (x - 4 + i)$ .

Expand $(x - 3) (x - 4 - i)$ by multiplying each term in the first expression by each term in the second expression.

$y = (x \cdot x + x \cdot - 4 + x (- i) - 3 x - 3 \cdot - 4 - 3 (- i)) (x - 4 + i)$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot - 4 + x (- i) - 3 x - 3 \cdot - 4 - 3 (- i)) (x - 4 + i)$

Move $- 4$ to the left of $x$ .

$y = (x^{2} - 4 \cdot x + x (- i) - 3 x - 3 \cdot - 4 - 3 (- i)) (x - 4 + i)$

Multiply $- 3$ by $- 4$ .

$y = (x^{2} - 4 x + x (- i) - 3 x + 12 - 3 (- i)) (x - 4 + i)$

Multiply $- 1$ by $- 3$ .

$y = (x^{2} - 4 x + x (- i) - 3 x + 12 + 3 i) (x - 4 + i)$

Subtract $3 x$ from $- 4 x$ .

$y = (x^{2} + x (- i) - 7 x + 12 + 3 i) (x - 4 + i)$

Expand $(x^{2} + x (- i) - 7 x + 12 + 3 i) (x - 4 + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot - 4 + x^{2} i + x (- i) x + x (- i) \cdot - 4 + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

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 $1$ .

$y = x^{2} x + x^{2} \cdot - 4 + x^{2} i + x (- i) x + x (- i) \cdot - 4 + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

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

$y = x^{2 + 1} + x^{2} \cdot - 4 + x^{2} i + x (- i) x + x (- i) \cdot - 4 + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Add $2$ and $1$ .

$y = x^{3} + x^{2} \cdot - 4 + x^{2} i + x (- i) x + x (- i) \cdot - 4 + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

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

$y = x^{3} - 4 \cdot x^{2} + x^{2} i + x (- i) x + x (- i) \cdot - 4 + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

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

Move $x$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x \cdot x (- i) + x (- i) \cdot - 4 + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $x$ by $x$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + x (- i) \cdot - 4 + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $- 4$ by $- 1$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + x (4 i) + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Move $4$ to the left of $x$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 \cdot (x i) + x (- i) i - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $x (- i) i$ .

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

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x (- (i i)) - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

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

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x (- (i i)) - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

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

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x (- i^{1 + 1}) - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Add $1$ and $1$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x (- i^{2}) - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

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

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x (1) - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $- 1$ by $- 1$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x \cdot 1 - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $x$ by $1$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x \cdot x - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

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

Move $x$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 (x \cdot x) - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $x$ by $x$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} - 7 x \cdot - 4 - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $- 4$ by $- 7$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x + 12 \cdot - 4 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $12$ by $- 4$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x + 3 i \cdot - 4 + 3 i i$

Multiply $- 4$ by $3$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i + 3 i i$

Multiply $3 i i$ .

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

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i + 3 (i i)$

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

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i + 3 (i i)$

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

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i + 3 i^{1 + 1}$

Add $1$ and $1$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i + 3 i^{2}$

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

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i + 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$y = x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i - 3$

Simplify by adding terms.

Combine the opposite terms in $x^{3} - 4 x^{2} + x^{2} i + x^{2} (- i) + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i - 3$ .

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

$y = x^{3} - 4 x^{2} + i x^{2} - i x^{2} + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i - 3$

Subtract $i x^{2}$ from $i x^{2}$ .

$y = x^{3} - 4 x^{2} + 0 + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i - 3$

Add $x^{3} - 4 x^{2}$ and $0$ .

$y = x^{3} - 4 x^{2} + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 12 i + 3 i x - 12 i - 3$

Subtract $12 i$ from $12 i$ .

$y = x^{3} - 4 x^{2} + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 3 i x + 0 - 3$

Add $x^{3} - 4 x^{2} + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 3 i x$ and $0$ .

$y = x^{3} - 4 x^{2} + 4 x i + x - 7 x^{2} + 28 x - 7 x i + 12 x - 48 + 3 i x - 3$

Subtract $7 x^{2}$ from $- 4 x^{2}$ .

$y = x^{3} - 11 x^{2} + 4 x i + x + 28 x - 7 x i + 12 x - 48 + 3 i x - 3$

Subtract $7 x i$ from $4 x i$ .

$y = x^{3} - 11 x^{2} - 3 x i + x + 28 x + 12 x - 48 + 3 i x - 3$

Combine the opposite terms in $x^{3} - 11 x^{2} - 3 x i + x + 28 x + 12 x - 48 + 3 i x - 3$ .

Reorder the factors in the terms $- 3 x i$ and $3 i x$ .

$y = x^{3} - 11 x^{2} - 3 i x + x + 28 x + 12 x - 48 + 3 i x - 3$

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

$y = x^{3} - 11 x^{2} + x + 28 x + 12 x - 48 + 0 - 3$

Add $x^{3} - 11 x^{2} + x + 28 x + 12 x - 48$ and $0$ .

$y = x^{3} - 11 x^{2} + x + 28 x + 12 x - 48 - 3$

Add $x$ and $28 x$ .

$y = x^{3} - 11 x^{2} + 29 x + 12 x - 48 - 3$

Add $29 x$ and $12 x$ .

$y = x^{3} - 11 x^{2} + 41 x - 48 - 3$

Subtract $3$ from $- 48$ .

$y = x^{3} - 11 x^{2} + 41 x - 51$

Step 7

Find the Equation Given the Roots 3 , 4+i

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