Algebra Examples

$1$ and $- 4 i$

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

The factor is $x - 1$

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.

$y = (x - 1) (x + 4 i) (x - 4 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 1) (x + 4 i) (x - 4 i)$ .

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

Apply the distributive property.

$y = (x (x + 4 i) - 1 (x + 4 i)) (x - 4 i)$

Step 6.1.2

Apply the distributive property.

$y = (x \cdot x + x (4 i) - 1 (x + 4 i)) (x - 4 i)$

Step 6.1.3

Apply the distributive property.

$y = (x \cdot x + x (4 i) - 1 x - 1 (4 i)) (x - 4 i)$

Step 6.2

Simplify each term.

Tap for more steps...

Step 6.2.1

Multiply $x$ by $x$ .

$y = (x^{2} + x (4 i) - 1 x - 1 (4 i)) (x - 4 i)$

Step 6.2.2

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

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

Step 6.2.3

Rewrite $- 1 x$ as $- x$ .

$y = (x^{2} + 4 x i - x - 1 (4 i)) (x - 4 i)$

Step 6.2.4

Multiply $4$ by $- 1$ .

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

Step 6.3

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

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

Step 6.4

Simplify terms.

Tap for more steps...

Step 6.4.1

Simplify each term.

Tap for more steps...

Step 6.4.1.1

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

Tap for more steps...

Step 6.4.1.1.1

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

Tap for more steps...

Step 6.4.1.1.1.1

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

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

Step 6.4.1.1.1.2

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

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

Step 6.4.1.1.2

Add $2$ and $1$ .

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

Step 6.4.1.2

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

Tap for more steps...

Step 6.4.1.2.1

Move $x$ .

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

Step 6.4.1.2.2

Multiply $x$ by $x$ .

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

Step 6.4.1.3

Multiply $4 x i (- 4 i)$ .

Tap for more steps...

Step 6.4.1.3.1

Multiply $- 4$ by $4$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i - 16 x i i - x \cdot x - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.3.2

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

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i - 16 x (i i) - x \cdot x - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.3.3

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

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i - 16 x (i i) - x \cdot x - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.3.4

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

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i - 16 x i^{1 + 1} - x \cdot x - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.3.5

Add $1$ and $1$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i - 16 x i^{2} - x \cdot x - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.4

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

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i - 16 x \cdot - 1 - x \cdot x - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.5

Multiply $- 1$ by $- 16$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x \cdot x - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.6

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

Tap for more steps...

Step 6.4.1.6.1

Move $x$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - (x \cdot x) - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.6.2

Multiply $x$ by $x$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} - x (- 4 i) - 4 i x - 4 i (- 4 i)$

Step 6.4.1.7

Multiply $- 4$ by $- 1$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x - 4 i (- 4 i)$

Step 6.4.1.8

Multiply $- 4 i (- 4 i)$ .

Tap for more steps...

Step 6.4.1.8.1

Multiply $- 4$ by $- 4$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x + 16 i i$

Step 6.4.1.8.2

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

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x + 16 (i i)$

Step 6.4.1.8.3

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

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x + 16 (i i)$

Step 6.4.1.8.4

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

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x + 16 i^{1 + 1}$

Step 6.4.1.8.5

Add $1$ and $1$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x + 16 i^{2}$

Step 6.4.1.9

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

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x + 16 \cdot - 1$

Step 6.4.1.10

Multiply $16$ by $- 1$ .

$y = x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x - 16$

Step 6.4.2

Simplify by adding terms.

Tap for more steps...

Step 6.4.2.1

Combine the opposite terms in $x^{3} + x^{2} (- 4 i) + 4 x^{2} i + 16 x - x^{2} + 4 x i - 4 i x - 16$ .

Tap for more steps...

Step 6.4.2.1.1

Reorder the factors in the terms $x^{2} (- 4 i)$ and $4 x^{2} i$ .

$y = x^{3} - 4 i x^{2} + 4 i x^{2} + 16 x - x^{2} + 4 x i - 4 i x - 16$

Step 6.4.2.1.2

Add $- 4 i x^{2}$ and $4 i x^{2}$ .

$y = x^{3} + 0 + 16 x - x^{2} + 4 x i - 4 i x - 16$

Step 6.4.2.1.3

Add $x^{3}$ and $0$ .

$y = x^{3} + 16 x - x^{2} + 4 x i - 4 i x - 16$

Step 6.4.2.1.4

Reorder the factors in the terms $4 x i$ and $- 4 i x$ .

$y = x^{3} + 16 x - x^{2} + 4 i x - 4 i x - 16$

Step 6.4.2.1.5

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

$y = x^{3} + 16 x - x^{2} + 0 - 16$

Step 6.4.2.1.6

Add $x^{3} + 16 x - x^{2}$ and $0$ .

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

Step 6.4.2.2

Move $16 x$ .

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

Step 7