Algebra Examples

$f (x) = (x + 2) (x - 1) (x - (4 + 3 i)) (x - (4 - 3 i))$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = (x + 2) (x - 1) (x - (4 + 3 i)) (x - (4 - 3 i))$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x + 2) (x - 1) (x - (4 + 3 i)) (x - (4 - 3 i)) = 0$ .

$(x + 2) (x - 1) (x - (4 + 3 i)) (x - (4 - 3 i)) = 0$

Step 1.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$x - 1 = 0$

$x - (4 + 3 i) = 0$

$x - (4 - 3 i) = 0$

Step 1.2.3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 1.2.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.4

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.4.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.5

Set $x - (4 + 3 i)$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - (4 + 3 i)$ equal to $0$ .

$x - (4 + 3 i) = 0$

Step 1.2.5.2

Solve $x - (4 + 3 i) = 0$ for $x$ .

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Apply the distributive property.

$x - 1 \cdot 4 - (3 i) = 0$

Step 1.2.5.2.1.2

Multiply $- 1$ by $4$ .

$x - 4 - (3 i) = 0$

Step 1.2.5.2.1.3

Multiply $3$ by $- 1$ .

$x - 4 - 3 i = 0$

Step 1.2.5.2.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.2.2.1

Add $4$ to both sides of the equation.

$x - 3 i = 4$

Step 1.2.5.2.2.2

Add $3 i$ to both sides of the equation.

$x = 4 + 3 i$

Step 1.2.6

Set $x - (4 - 3 i)$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x - (4 - 3 i)$ equal to $0$ .

$x - (4 - 3 i) = 0$

Step 1.2.6.2

Solve $x - (4 - 3 i) = 0$ for $x$ .

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Step 1.2.6.2.1

Simplify each term.

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Step 1.2.6.2.1.1

Apply the distributive property.

$x - 1 \cdot 4 - (- 3 i) = 0$

Step 1.2.6.2.1.2

Multiply $- 1$ by $4$ .

$x - 4 - (- 3 i) = 0$

Step 1.2.6.2.1.3

Multiply $- 3$ by $- 1$ .

$x - 4 + 3 i = 0$

Step 1.2.6.2.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.6.2.2.1

Add $4$ to both sides of the equation.

$x + 3 i = 4$

Step 1.2.6.2.2.2

Subtract $3 i$ from both sides of the equation.

$x = 4 - 3 i$

Step 1.2.7

The final solution is all the values that make $(x + 2) (x - 1) (x - (4 + 3 i)) (x - (4 - 3 i)) = 0$ true.

$x = - 2, 1, 4 + 3 i, 4 - 3 i$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 2, 0), (1, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = ((0) + 2) ((0) - 1) ((0) - (4 + 3 i)) ((0) - (4 - 3 i))$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = (0 + 2) ((0) - 1) ((0) - (4 + 3 i)) ((0) - (4 - 3 i))$

Step 2.2.2

Remove parentheses.

$y = (0 + 2) (0 - 1) ((0) - (4 + 3 i)) ((0) - (4 - 3 i))$

Step 2.2.3

Remove parentheses.

$y = (0 + 2) (0 - 1) (0 - (4 + 3 i)) ((0) - (4 - 3 i))$

Step 2.2.4

Remove parentheses.

$y = (0 + 2) (0 - 1) (0 - (4 + 3 i)) (0 - (4 - 3 i))$

Step 2.2.5

Remove parentheses.

$y = ((0) + 2) ((0) - 1) ((0) - (4 + 3 i)) ((0) - (4 - 3 i))$

Step 2.2.6

Simplify $((0) + 2) ((0) - 1) ((0) - (4 + 3 i)) ((0) - (4 - 3 i))$ .

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Step 2.2.6.1

Simplify terms.

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Step 2.2.6.1.1

Add $0$ and $2$ .

$y = 2 ((0) - 1) ((0) - (4 + 3 i)) ((0) - (4 - 3 i))$

Step 2.2.6.1.2

Simplify the expression.

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Step 2.2.6.1.2.1

Subtract $1$ from $0$ .

$y = 2 \cdot - 1 ((0) - (4 + 3 i)) ((0) - (4 - 3 i))$

Step 2.2.6.1.2.2

Multiply $2$ by $- 1$ .

$y = - 2 ((0) - (4 + 3 i)) ((0) - (4 - 3 i))$

Step 2.2.6.1.3

Simplify each term.

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Step 2.2.6.1.3.1

Apply the distributive property.

$y = - 2 (0 - 1 \cdot 4 - (3 i)) ((0) - (4 - 3 i))$

Step 2.2.6.1.3.2

Multiply $- 1$ by $4$ .

$y = - 2 (0 - 4 - (3 i)) ((0) - (4 - 3 i))$

Step 2.2.6.1.3.3

Multiply $3$ by $- 1$ .

$y = - 2 (0 - 4 - 3 i) ((0) - (4 - 3 i))$

Step 2.2.6.1.4

Simplify by multiplying through.

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Step 2.2.6.1.4.1

Subtract $4$ from $0$ .

$y = - 2 (- 4 - 3 i) ((0) - (4 - 3 i))$

Step 2.2.6.1.4.2

Apply the distributive property.

$y = (- 2 \cdot - 4 - 2 (- 3 i)) ((0) - (4 - 3 i))$

Step 2.2.6.1.4.3

Multiply.

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Step 2.2.6.1.4.3.1

Multiply $- 2$ by $- 4$ .

$y = (8 - 2 (- 3 i)) ((0) - (4 - 3 i))$

Step 2.2.6.1.4.3.2

Multiply $- 3$ by $- 2$ .

$y = (8 + 6 i) ((0) - (4 - 3 i))$

Step 2.2.6.1.5

Simplify each term.

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Step 2.2.6.1.5.1

Apply the distributive property.

$y = (8 + 6 i) (0 - 1 \cdot 4 - (- 3 i))$

Step 2.2.6.1.5.2

Multiply $- 1$ by $4$ .

$y = (8 + 6 i) (0 - 4 - (- 3 i))$

Step 2.2.6.1.5.3

Multiply $- 3$ by $- 1$ .

$y = (8 + 6 i) (0 - 4 + 3 i)$

Step 2.2.6.1.6

Subtract $4$ from $0$ .

$y = (8 + 6 i) (- 4 + 3 i)$

Step 2.2.6.2

Expand $(8 + 6 i) (- 4 + 3 i)$ using the FOIL Method.

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Step 2.2.6.2.1

Apply the distributive property.

$y = 8 (- 4 + 3 i) + 6 i (- 4 + 3 i)$

Step 2.2.6.2.2

Apply the distributive property.

$y = 8 \cdot - 4 + 8 (3 i) + 6 i (- 4 + 3 i)$

Step 2.2.6.2.3

Apply the distributive property.

$y = 8 \cdot - 4 + 8 (3 i) + 6 i \cdot - 4 + 6 i (3 i)$

Step 2.2.6.3

Simplify and combine like terms.

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Step 2.2.6.3.1

Simplify each term.

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Step 2.2.6.3.1.1

Multiply $8$ by $- 4$ .

$y = - 32 + 8 (3 i) + 6 i \cdot - 4 + 6 i (3 i)$

Step 2.2.6.3.1.2

Multiply $3$ by $8$ .

$y = - 32 + 24 i + 6 i \cdot - 4 + 6 i (3 i)$

Step 2.2.6.3.1.3

Multiply $- 4$ by $6$ .

$y = - 32 + 24 i - 24 i + 6 i (3 i)$

Step 2.2.6.3.1.4

Multiply $6 i (3 i)$ .

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Step 2.2.6.3.1.4.1

Multiply $3$ by $6$ .

$y = - 32 + 24 i - 24 i + 18 i i$

Step 2.2.6.3.1.4.2

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

$y = - 32 + 24 i - 24 i + 18 (i^{1} i)$

Step 2.2.6.3.1.4.3

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

$y = - 32 + 24 i - 24 i + 18 (i^{1} i^{1})$

Step 2.2.6.3.1.4.4

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

$y = - 32 + 24 i - 24 i + 18 i^{1 + 1}$

Step 2.2.6.3.1.4.5

Add $1$ and $1$ .

$y = - 32 + 24 i - 24 i + 18 i^{2}$

Step 2.2.6.3.1.5

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

$y = - 32 + 24 i - 24 i + 18 \cdot - 1$

Step 2.2.6.3.1.6

Multiply $18$ by $- 1$ .

$y = - 32 + 24 i - 24 i - 18$

Step 2.2.6.3.2

Subtract $18$ from $- 32$ .

$y = - 50 + 24 i - 24 i$

Step 2.2.6.3.3

Subtract $24 i$ from $24 i$ .

$y = - 50 + 0$

Step 2.2.6.3.4

Add $- 50$ and $0$ .

$y = - 50$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 50)$

Step 3

List the intersections.

x-intercept(s): $(- 2, 0), (1, 0)$

y-intercept(s): $(0, - 50)$

Step 4