Algebra Examples

$p (x) = 2 x^{3} + 5 x^{2} + x - 2$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = 2 {(- 2)}^{3} + 5 {(- 2)}^{2} - 2 - 2$

Step 1.2

Simplify the result.

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Step 1.2.1

Remove parentheses.

$f (- 2) = 2 {(- 2)}^{3} + 5 {(- 2)}^{2} - 2 - 2$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Raise $- 2$ to the power of $3$ .

$f (- 2) = 2 \cdot - 8 + 5 {(- 2)}^{2} - 2 - 2$

Step 1.2.2.2

Multiply $2$ by $- 8$ .

$f (- 2) = - 16 + 5 {(- 2)}^{2} - 2 - 2$

Step 1.2.2.3

Raise $- 2$ to the power of $2$ .

$f (- 2) = - 16 + 5 \cdot 4 - 2 - 2$

Step 1.2.2.4

Multiply $5$ by $4$ .

$f (- 2) = - 16 + 20 - 2 - 2$

Step 1.2.3

Simplify by adding and subtracting.

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Step 1.2.3.1

Add $- 16$ and $20$ .

$f (- 2) = 4 - 2 - 2$

Step 1.2.3.2

Subtract $2$ from $4$ .

$f (- 2) = 2 - 2$

Step 1.2.3.3

Subtract $2$ from $2$ .

$f (- 2) = 0$

Step 1.2.4

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 0$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 2 {(0)}^{3} + 5 {(0)}^{2} + 0 - 2$

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

$f (0) = 2 {(0)}^{3} + 5 {(0)}^{2} + 0 - 2$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 2 \cdot 0 + 5 {(0)}^{2} + 0 - 2$

Step 2.2.2.2

Multiply $2$ by $0$ .

$f (0) = 0 + 5 {(0)}^{2} + 0 - 2$

Step 2.2.2.3

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 5 \cdot 0 + 0 - 2$

Step 2.2.2.4

Multiply $5$ by $0$ .

$f (0) = 0 + 0 + 0 - 2$

Step 2.2.3

Simplify by adding and subtracting.

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Step 2.2.3.1

Add $0$ and $0$ .

$f (0) = 0 + 0 - 2$

Step 2.2.3.2

Add $0$ and $0$ .

$f (0) = 0 - 2$

Step 2.2.3.3

Subtract $2$ from $0$ .

$f (0) = - 2$

Step 2.2.4

The final answer is $- 2$ .

$- 2$

Step 2.3

Convert $- 2$ to decimal.

$y = - 2$

Step 3

Find the point at $x = 1$ .

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 2 {(1)}^{3} + 5 {(1)}^{2} + 1 - 2$

Step 3.2

Simplify the result.

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Step 3.2.1

Remove parentheses.

$f (1) = 2 {(1)}^{3} + 5 {(1)}^{2} + 1 - 2$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

One to any power is one.

$f (1) = 2 \cdot 1 + 5 {(1)}^{2} + 1 - 2$

Step 3.2.2.2

Multiply $2$ by $1$ .

$f (1) = 2 + 5 {(1)}^{2} + 1 - 2$

Step 3.2.2.3

One to any power is one.

$f (1) = 2 + 5 \cdot 1 + 1 - 2$

Step 3.2.2.4

Multiply $5$ by $1$ .

$f (1) = 2 + 5 + 1 - 2$

Step 3.2.3

Simplify by adding and subtracting.

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Step 3.2.3.1

Add $2$ and $5$ .

$f (1) = 7 + 1 - 2$

Step 3.2.3.2

Add $7$ and $1$ .

$f (1) = 8 - 2$

Step 3.2.3.3

Subtract $2$ from $8$ .

$f (1) = 6$

Step 3.2.4

The final answer is $6$ .

$6$

Step 3.3

Convert $6$ to decimal.

$y = 6$

Step 4

Find the point at $x = 2$ .

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Step 4.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 2 {(2)}^{3} + 5 {(2)}^{2} + 2 - 2$

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

$f (2) = 2 {(2)}^{3} + 5 {(2)}^{2} + 2 - 2$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Multiply $2$ by ${(2)}^{3}$ by adding the exponents.

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Step 4.2.2.1.1

Multiply $2$ by ${(2)}^{3}$ .

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Step 4.2.2.1.1.1

Raise $2$ to the power of $1$ .

$f (2) = 2 {(2)}^{3} + 5 {(2)}^{2} + 2 - 2$

Step 4.2.2.1.1.2

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

$f (2) = 2^{1 + 3} + 5 {(2)}^{2} + 2 - 2$

Step 4.2.2.1.2

Add $1$ and $3$ .

$f (2) = 2^{4} + 5 {(2)}^{2} + 2 - 2$

Step 4.2.2.2

Raise $2$ to the power of $4$ .

$f (2) = 16 + 5 {(2)}^{2} + 2 - 2$

Step 4.2.2.3

Raise $2$ to the power of $2$ .

$f (2) = 16 + 5 \cdot 4 + 2 - 2$

Step 4.2.2.4

Multiply $5$ by $4$ .

$f (2) = 16 + 20 + 2 - 2$

Step 4.2.3

Simplify by adding and subtracting.

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Step 4.2.3.1

Add $16$ and $20$ .

$f (2) = 36 + 2 - 2$

Step 4.2.3.2

Add $36$ and $2$ .

$f (2) = 38 - 2$

Step 4.2.3.3

Subtract $2$ from $38$ .

$f (2) = 36$

Step 4.2.4

The final answer is $36$ .

$36$

Step 4.3

Convert $36$ to decimal.

$y = 36$

Step 5

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 0 \\ 0 & - 2 \\ 1 & 6 \\ 2 & 36 \end{array}$

Step 6

The cubic function can be graphed using the function behavior and the selected points.

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 0 \\ 0 & - 2 \\ 1 & 6 \\ 2 & 36 \end{array}$

Step 7