Algebra Examples

$f (x) = (x + 4) (x - 5)$

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

Expand $(x + 4) (x - 5)$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$x (x - 5) + 4 (x - 5)$

Step 1.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 5 + 4 (x - 5)$

Step 1.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 5 + 4 x + 4 \cdot - 5$

Step 1.1.2

Simplify and combine like terms.

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Step 1.1.2.1

Simplify each term.

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Step 1.1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 5 + 4 x + 4 \cdot - 5$

Step 1.1.2.1.2

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x + 4 x + 4 \cdot - 5$

Step 1.1.2.1.3

Multiply $4$ by $- 5$ .

$x^{2} - 5 x + 4 x - 20$

Step 1.1.2.2

Add $- 5 x$ and $4 x$ .

$x^{2} - x - 20$

Step 1.2

The largest exponent is the degree of the polynomial.

$2$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

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

Expand $(x + 4) (x - 5)$ using the FOIL Method.

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Step 3.1.1.1

Apply the distributive property.

$x (x - 5) + 4 (x - 5)$

Step 3.1.1.2

Apply the distributive property.

$x \cdot x + x \cdot - 5 + 4 (x - 5)$

Step 3.1.1.3

Apply the distributive property.

$x \cdot x + x \cdot - 5 + 4 x + 4 \cdot - 5$

Step 3.1.2

Simplify and combine like terms.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 5 + 4 x + 4 \cdot - 5$

Step 3.1.2.1.2

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x + 4 x + 4 \cdot - 5$

Step 3.1.2.1.3

Multiply $4$ by $- 5$ .

$x^{2} - 5 x + 4 x - 20$

Step 3.1.2.2

Add $- 5 x$ and $4 x$ .

$x^{2} - x - 20$

Step 3.2

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

$x^{2}$

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.

Rises to the left and rises to the right

Step 7