Algebra Examples

y = x^{3} + x^{2} + x + 1

$y = x^{3} + x^{2} + x + 1$

Step 1

Find the point at

x = - 2

$x = - 2$ .

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Step 1.1

Replace the variable

x

$x$ with

- 2

$- 2$ in the expression.

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Remove parentheses.

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

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Raise

- 2

$- 2$ to the power of

3

$3$ .

f (- 2) = - 8 + {(- 2)}^{2} - 2 + 1

$f (- 2) = - 8 + {(- 2)}^{2} - 2 + 1$

Step 1.2.2.2

Raise

- 2

$- 2$ to the power of

2

$2$ .

f (- 2) = - 8 + 4 - 2 + 1

$f (- 2) = - 8 + 4 - 2 + 1$

f (- 2) = - 8 + 4 - 2 + 1

$f (- 2) = - 8 + 4 - 2 + 1$

Step 1.2.3

Simplify by adding and subtracting.

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Step 1.2.3.1

Add

- 8

$- 8$ and

4

$4$ .

f (- 2) = - 4 - 2 + 1

$f (- 2) = - 4 - 2 + 1$

Step 1.2.3.2

Subtract

2

$2$ from

- 4

$- 4$ .

f (- 2) = - 6 + 1

$f (- 2) = - 6 + 1$

Step 1.2.3.3

Add

- 6

$- 6$ and

1

$1$ .

f (- 2) = - 5

$f (- 2) = - 5$

f (- 2) = - 5

$f (- 2) = - 5$

Step 1.2.4

The final answer is

- 5

$- 5$ .

- 5

$- 5$

- 5

$- 5$

Step 1.3

Convert

- 5

$- 5$ to decimal.

y = - 5

$y = - 5$

y = - 5

$y = - 5$

Step 2

Find the point at

x = - 1

$x = - 1$ .

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Step 2.1

Replace the variable

x

$x$ with

- 1

$- 1$ in the expression.

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

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

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Raise

- 1

$- 1$ to the power of

3

$3$ .

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

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

Step 2.2.2.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

f (- 1) = - 1 + 1 - 1 + 1

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

f (- 1) = - 1 + 1 - 1 + 1

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

Step 2.2.3

Simplify by adding and subtracting.

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Step 2.2.3.1

Add

- 1

$- 1$ and

1

$1$ .

f (- 1) = 0 - 1 + 1

$f (- 1) = 0 - 1 + 1$

Step 2.2.3.2

Subtract

1

$1$ from

0

$0$ .

f (- 1) = - 1 + 1

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

Step 2.2.3.3

Add

- 1

$- 1$ and

1

$1$ .

f (- 1) = 0

$f (- 1) = 0$

f (- 1) = 0

$f (- 1) = 0$

Step 2.2.4

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 2.3

Convert

0

$0$ to decimal.

y = 0

$y = 0$

y = 0

$y = 0$

Step 3

Find the point at

x = 0

$x = 0$ .

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Step 3.1

Replace the variable

x

$x$ with

0

$0$ in the expression.

f (0) = {(0)}^{3} + {(0)}^{2} + 0 + 1

$f (0) = {(0)}^{3} + {(0)}^{2} + 0 + 1$

Step 3.2

Simplify the result.

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Step 3.2.1

Remove parentheses.

$f (0) = {(0)}^{3} + {(0)}^{2} + 0 + 1$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Raising

$0$ to any positive power yields

$0$ .

$f (0) = 0 + {(0)}^{2} + 0 + 1$

Step 3.2.2.2

Raising

$0$ to any positive power yields

$0$ .

$f (0) = 0 + 0 + 0 + 1$

Step 3.2.3

Simplify by adding numbers.

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Step 3.2.3.1

Add

$0$ and

$0$ .

$f (0) = 0 + 0 + 1$

Step 3.2.3.2

Add

$0$ and

$0$ .

$f (0) = 0 + 1$

Step 3.2.3.3

Add

$0$ and

$1$ .

$f (0) = 1$

Step 3.2.4

The final answer is

$1$ .

$1$

Step 3.3

Convert

$1$ to decimal.

$y = 1$

Step 4

Find the point at

$x = 1$ .

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Step 4.1

Replace the variable

$x$ with

$1$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

One to any power is one.

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

Step 4.2.2.2

One to any power is one.

$f (1) = 1 + 1 + 1 + 1$

Step 4.2.3

Simplify by adding numbers.

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Step 4.2.3.1

Add

$1$ and

$1$ .

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

Step 4.2.3.2

Add

$2$ and

$1$ .

$f (1) = 3 + 1$

Step 4.2.3.3

Add

$3$ and

$1$ .

$f (1) = 4$

Step 4.2.4

The final answer is

$4$ .

$4$

Step 4.3

Convert

$4$ to decimal.

$y = 4$

Step 5

Find the point at

$x = 2$ .

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Step 5.1

Replace the variable

$x$ with

$2$ in the expression.

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

Step 5.2

Simplify the result.

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Step 5.2.1

Remove parentheses.

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

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Raise

$2$ to the power of

$3$ .

$f (2) = 8 + {(2)}^{2} + 2 + 1$

Step 5.2.2.2

Raise

$2$ to the power of

$2$ .

$f (2) = 8 + 4 + 2 + 1$

Step 5.2.3

Simplify by adding numbers.

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Step 5.2.3.1

Add

$8$ and

$4$ .

$f (2) = 12 + 2 + 1$

Step 5.2.3.2

Add

$12$ and

$2$ .

$f (2) = 14 + 1$

Step 5.2.3.3

Add

$14$ and

$1$ .

$f (2) = 15$

Step 5.2.4

The final answer is

$15$ .

$15$

Step 5.3

Convert

$15$ to decimal.

$y = 15$

Step 6

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

$\begin{array}{c} x & y \\ - 2 & - 5 \\ - 1 & 0 \\ 0 & 1 \\ 1 & 4 \\ 2 & 15 \end{array}$

Step 7

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & - 5 \\ - 1 & 0 \\ 0 & 1 \\ 1 & 4 \\ 2 & 15 \end{array}$

Step 8