Algebra Examples

5 + 4 i

$5 + 4 i$ ,

5 - 4 i

$5 - 4 i$ ,

- 1

$- 1$

Step 1

Roots are the points where the graph intercepts with the x-axis

(y = 0)

$(y = 0)$ .

y = 0

$y = 0$ at the roots

Step 2

The root at

x = 5 + 4 i

$x = 5 + 4 i$ was found by solving for

x

$x$ when

x - (5 + 4 i) = y

$x - (5 + 4 i) = y$ and

y = 0

$y = 0$ .

The factor is

x - 5 - 4 i

$x - 5 - 4 i$

Step 3

The root at

x = 5 - 4 i

$x = 5 - 4 i$ was found by solving for

x

$x$ when

x - (5 - 4 i) = y

$x - (5 - 4 i) = y$ and

y = 0

$y = 0$ .

The factor is

x - 5 + 4 i

$x - 5 + 4 i$

Step 4

The root at

x = - 1

$x = - 1$ was found by solving for

x

$x$ when

x - (- 1) = y

$x - (- 1) = y$ and

y = 0

$y = 0$ .

The factor is

x + 1

$x + 1$

Step 5

Combine all the factors into a single equation.

y = (x - 5 - 4 i) (x - 5 + 4 i) (x + 1)

$y = (x - 5 - 4 i) (x - 5 + 4 i) (x + 1)$

Step 6

Multiply all the factors to simplify the equation

y = (x - 5 - 4 i) (x - 5 + 4 i) (x + 1)

$y = (x - 5 - 4 i) (x - 5 + 4 i) (x + 1)$ .

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Step 6.1

Expand

(x - 5 - 4 i) (x - 5 + 4 i)

$(x - 5 - 4 i) (x - 5 + 4 i)$ by multiplying each term in the first expression by each term in the second expression.

y = (x \cdot x + x \cdot - 5 + x (4 i) - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)

$y = (x \cdot x + x \cdot - 5 + x (4 i) - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

Step 6.2

Simplify terms.

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Step 6.2.1

Combine the opposite terms in

x \cdot x + x \cdot - 5 + x (4 i) - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)

$x \cdot x + x \cdot - 5 + x (4 i) - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)$ .

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Step 6.2.1.1

Reorder the factors in the terms

x (4 i)

$x (4 i)$ and

- 4 i x

$- 4 i x$ .

y = (x \cdot x + x \cdot - 5 + 4 i x - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)

$y = (x \cdot x + x \cdot - 5 + 4 i x - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i x - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

Step 6.2.1.2

Subtract

4 i x

$4 i x$ from

4 i x

$4 i x$ .

y = (x \cdot x + x \cdot - 5 + 0 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)

$y = (x \cdot x + x \cdot - 5 + 0 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

Step 6.2.1.3

Add

x \cdot x + x \cdot - 5

$x \cdot x + x \cdot - 5$ and

0

$0$ .

y = (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)

$y = (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

y = (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)

$y = (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

Multiply

x

$x$ by

x

$x$ .

y = (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)

$y = (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

Step 6.2.2.2

Move

$- 5$ to the left of

$x$ .

$y = (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

Step 6.2.2.3

Multiply

$- 5$ by

$- 5$ .

$y = (x^{2} - 5 x - 5 x + 25 - 5 (4 i) - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

Step 6.2.2.4

Multiply

$4$ by

$- 5$ .

$y = (x^{2} - 5 x - 5 x + 25 - 20 i - 4 i \cdot - 5 - 4 i (4 i)) (x + 1)$

Step 6.2.2.5

Multiply

$- 5$ by

$- 4$ .

$y = (x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 4 i (4 i)) (x + 1)$

Step 6.2.2.6

Multiply

$- 4 i (4 i)$ .

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Step 6.2.2.6.1

Multiply

$4$ by

$- 4$ .

$y = (x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 i i) (x + 1)$

Step 6.2.2.6.2

Raise

$i$ to the power of

$1$ .

$y = (x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 (i i)) (x + 1)$

Step 6.2.2.6.3

Raise

$i$ to the power of

$1$ .

$y = (x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 (i i)) (x + 1)$

Step 6.2.2.6.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = (x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 i^{1 + 1}) (x + 1)$

Step 6.2.2.6.5

Add

$1$ and

$1$ .

$y = (x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 i^{2}) (x + 1)$

Step 6.2.2.7

Rewrite

$i^{2}$ as

$- 1$ .

$y = (x^{2} - 5 x - 5 x + 25 - 20 i + 20 i - 16 \cdot - 1) (x + 1)$

Step 6.2.2.8

Multiply

$- 16$ by

$- 1$ .

$y = (x^{2} - 5 x - 5 x + 25 - 20 i + 20 i + 16) (x + 1)$

Step 6.2.3

Simplify by adding terms.

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Step 6.2.3.1

Combine the opposite terms in

$x^{2} - 5 x - 5 x + 25 - 20 i + 20 i + 16$ .

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Step 6.2.3.1.1

Add

$- 20 i$ and

$20 i$ .

$y = (x^{2} - 5 x - 5 x + 25 + 0 + 16) (x + 1)$

Step 6.2.3.1.2

Add

$x^{2} - 5 x - 5 x + 25$ and

$0$ .

$y = (x^{2} - 5 x - 5 x + 25 + 16) (x + 1)$

Step 6.2.3.2

Subtract

$5 x$ from

$- 5 x$ .

$y = (x^{2} - 10 x + 25 + 16) (x + 1)$

Step 6.2.3.3

Add

$25$ and

$16$ .

$y = (x^{2} - 10 x + 41) (x + 1)$

Step 6.3

Expand

$(x^{2} - 10 x + 41) (x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} \cdot 1 - 10 x \cdot x - 10 x \cdot 1 + 41 x + 41 \cdot 1$

Step 6.4

Simplify terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply

$x^{2}$ by

$x$ by adding the exponents.

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Step 6.4.1.1.1

Multiply

$x^{2}$ by

$x$ .

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Step 6.4.1.1.1.1

Raise

$x$ to the power of

$1$ .

$y = x^{2} x + x^{2} \cdot 1 - 10 x \cdot x - 10 x \cdot 1 + 41 x + 41 \cdot 1$

Step 6.4.1.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{2 + 1} + x^{2} \cdot 1 - 10 x \cdot x - 10 x \cdot 1 + 41 x + 41 \cdot 1$

Step 6.4.1.1.2

Add

$2$ and

$1$ .

$y = x^{3} + x^{2} \cdot 1 - 10 x \cdot x - 10 x \cdot 1 + 41 x + 41 \cdot 1$

Step 6.4.1.2

Multiply

$x^{2}$ by

$1$ .

$y = x^{3} + x^{2} - 10 x \cdot x - 10 x \cdot 1 + 41 x + 41 \cdot 1$

Step 6.4.1.3

Multiply

$x$ by

$x$ by adding the exponents.

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Step 6.4.1.3.1

Move

$x$ .

$y = x^{3} + x^{2} - 10 (x \cdot x) - 10 x \cdot 1 + 41 x + 41 \cdot 1$

Step 6.4.1.3.2

Multiply

$x$ by

$x$ .

$y = x^{3} + x^{2} - 10 x^{2} - 10 x \cdot 1 + 41 x + 41 \cdot 1$

Step 6.4.1.4

Multiply

$- 10$ by

$1$ .

$y = x^{3} + x^{2} - 10 x^{2} - 10 x + 41 x + 41 \cdot 1$

Step 6.4.1.5

Multiply

$41$ by

$1$ .

$y = x^{3} + x^{2} - 10 x^{2} - 10 x + 41 x + 41$

Step 6.4.2

Simplify by adding terms.

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Step 6.4.2.1

Subtract

$10 x^{2}$ from

$x^{2}$ .

$y = x^{3} - 9 x^{2} - 10 x + 41 x + 41$

Step 6.4.2.2

Add

$- 10 x$ and

$41 x$ .

$y = x^{3} - 9 x^{2} + 31 x + 41$

Step 7