Algebra Examples

$f (x) = x^{2} (x - 5)$

Step 1

Identify the degree of the function.

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Step 1.1

Simplify and reorder the polynomial.

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Step 1.1.1

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 5$

Step 1.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.2.1

Multiply $x^{2}$ by $x$ .

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Step 1.1.2.1.1

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

$x^{2} x^{1} + x^{2} \cdot - 5$

Step 1.1.2.1.2

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

$x^{2 + 1} + x^{2} \cdot - 5$

Step 1.1.2.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot - 5$

Step 1.1.3

Move $- 5$ to the left of $x^{2}$ .

$x^{3} - 5 x^{2}$

Step 1.2

The largest exponent is the degree of the polynomial.

$3$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

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Step 3.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 3.1.1

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 5$

Step 3.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.1.2.1

Multiply $x^{2}$ by $x$ .

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Step 3.1.2.1.1

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

$x^{2} x^{1} + x^{2} \cdot - 5$

Step 3.1.2.1.2

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

$x^{2 + 1} + x^{2} \cdot - 5$

Step 3.1.2.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot - 5$

Step 3.1.3

Move $- 5$ to the left of $x^{2}$ .

$x^{3} - 5 x^{2}$

Step 3.2

The leading term in a polynomial is the term with the highest degree.

$x^{3}$

Step 3.3

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 4

Since the leading coefficient is positive, the graph rises to the right.

Positive

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Falls to the left and rises to the right

Step 7