Algebra Examples

$p (x) = (x - 2) (x + \frac{1}{2}) (x - 4)$

Step 1

Find the point at $x = 0$ .

Tap for more steps...

Step 1.1

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

$f (0) = \frac{(2 (0) + 1) ((0) - 4) ((0) - 2)}{2}$

Step 1.2

Simplify the result.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $(0) - 4$ and $2$ .

Tap for more steps...

Step 1.2.1.1

Factor $2$ out of $(2 (0) + 1) ((0) - 4) ((0) - 2)$ .

$f (0) = \frac{2 (((2 (0) + 1) (0 - 2)) ((0) - 2))}{2}$

Step 1.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.1.2.1

Factor $2$ out of $2$ .

$f (0) = \frac{2 (((2 (0) + 1) (0 - 2)) ((0) - 2))}{2 (1)}$

Step 1.2.1.2.2

Cancel the common factor.

$f (0) = \frac{2 (((2 (0) + 1) (0 - 2)) ((0) - 2))}{2 \cdot 1}$

Step 1.2.1.2.3

Rewrite the expression.

$f (0) = \frac{((2 (0) + 1) (0 - 2)) ((0) - 2)}{1}$

Step 1.2.1.2.4

Divide $((2 (0) + 1) (0 - 2)) ((0) - 2)$ by $1$ .

$f (0) = ((2 (0) + 1) (0 - 2)) ((0) - 2)$

Step 1.2.2

Simplify the expression.

Tap for more steps...

Step 1.2.2.1

Multiply $2$ by $0$ .

$f (0) = (0 + 1) (0 - 2) ((0) - 2)$

Step 1.2.2.2

Add $0$ and $1$ .

$f (0) = 1 (0 - 2) ((0) - 2)$

Step 1.2.2.3

Multiply $0 - 2$ by $1$ .

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

Step 1.2.2.4

Subtract $2$ from $0$ .

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

Step 1.2.2.5

Subtract $2$ from $0$ .

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

Step 1.2.2.6

Multiply $- 2$ by $- 2$ .

$f (0) = 4$

Step 1.2.3

The final answer is $4$ .

$4$

Step 1.3

Convert $4$ to decimal.

$y = 4$

Step 2

Find the point at $x = 2$ .

Tap for more steps...

Step 2.1

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

$f (2) = \frac{(2 (2) + 1) ((2) - 4) ((2) - 2)}{2}$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $(2) - 4$ and $2$ .

Tap for more steps...

Step 2.2.1.1

Factor $2$ out of $(2 (2) + 1) ((2) - 4) ((2) - 2)$ .

$f (2) = \frac{2 (((2 (2) + 1) (1 - 2)) ((2) - 2))}{2}$

Step 2.2.1.2

Cancel the common factors.

Tap for more steps...

Step 2.2.1.2.1

Factor $2$ out of $2$ .

$f (2) = \frac{2 (((2 (2) + 1) (1 - 2)) ((2) - 2))}{2 (1)}$

Step 2.2.1.2.2

Cancel the common factor.

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

Step 2.2.1.2.3

Rewrite the expression.

$f (2) = \frac{((2 (2) + 1) (1 - 2)) ((2) - 2)}{1}$

Step 2.2.1.2.4

Divide $((2 (2) + 1) (1 - 2)) ((2) - 2)$ by $1$ .

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

Step 2.2.2

Simplify the expression.

Tap for more steps...

Step 2.2.2.1

Multiply $2$ by $2$ .

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

Step 2.2.2.2

Add $4$ and $1$ .

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

Step 2.2.2.3

Subtract $2$ from $1$ .

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

Step 2.2.2.4

Multiply $5$ by $- 1$ .

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

Step 2.2.2.5

Subtract $2$ from $2$ .

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

Step 2.2.2.6

Multiply $- 5$ by $0$ .

$f (2) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 1$ .

Tap for more steps...

Step 3.1

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

$f (1) = \frac{(2 (1) + 1) ((1) - 4) ((1) - 2)}{2}$

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

Simplify the numerator.

Tap for more steps...

Step 3.2.1.1

Multiply $2$ by $1$ .

$f (1) = \frac{(2 + 1) (1 - 4) (1 - 2)}{2}$

Step 3.2.1.2

Add $2$ and $1$ .

$f (1) = \frac{3 (1 - 4) (1 - 2)}{2}$

Step 3.2.1.3

Subtract $4$ from $1$ .

$f (1) = \frac{3 \cdot (- 3 (1 - 2))}{2}$

Step 3.2.1.4

Multiply $3$ by $- 3$ .

$f (1) = \frac{- 9 (1 - 2)}{2}$

Step 3.2.1.5

Subtract $2$ from $1$ .

$f (1) = \frac{- 9 \cdot - 1}{2}$

Step 3.2.2

Multiply $- 9$ by $- 1$ .

$f (1) = \frac{9}{2}$

Step 3.2.3

The final answer is $\frac{9}{2}$ .

$\frac{9}{2}$

Step 3.3

Convert $\frac{9}{2}$ to decimal.

$y = 4.5$

Step 4

Find the point at $x = 3$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{(2 (3) + 1) ((3) - 4) ((3) - 2)}{2}$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

Simplify the numerator.

Tap for more steps...

Step 4.2.1.1

Multiply $2$ by $3$ .

$f (3) = \frac{(6 + 1) (3 - 4) (3 - 2)}{2}$

Step 4.2.1.2

Add $6$ and $1$ .

$f (3) = \frac{7 (3 - 4) (3 - 2)}{2}$

Step 4.2.1.3

Subtract $4$ from $3$ .

$f (3) = \frac{7 \cdot (- 1 (3 - 2))}{2}$

Step 4.2.1.4

Combine exponents.

Tap for more steps...

Step 4.2.1.4.1

Factor out negative.

$f (3) = \frac{- (7 (3 - 2))}{2}$

Step 4.2.1.4.2

Multiply $7$ by $- 1$ .

$f (3) = \frac{- 7 (3 - 2)}{2}$

Step 4.2.1.5

Subtract $2$ from $3$ .

$f (3) = \frac{- 7 \cdot 1}{2}$

Step 4.2.2

Simplify the expression.

Tap for more steps...

Step 4.2.2.1

Multiply $- 7$ by $1$ .

$f (3) = \frac{- 7}{2}$

Step 4.2.2.2

Move the negative in front of the fraction.

$f (3) = - \frac{7}{2}$

Step 4.2.3

The final answer is $- \frac{7}{2}$ .

$- \frac{7}{2}$

Step 4.3

Convert $- \frac{7}{2}$ to decimal.

$y = - 3.5$

Step 5

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

$\begin{array}{c} x & y \\ 0 & 4 \\ 1 & 4.5 \\ 2 & 0 \\ 3 & - 3.5 \\ 4 & 0 \end{array}$

Step 6

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

Rises to the left and rises to the right

$\begin{array}{c} x & y \\ 0 & 4 \\ 1 & 4.5 \\ 2 & 0 \\ 3 & - 3.5 \\ 4 & 0 \end{array}$

Step 7