Algebra Examples

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

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

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$5 x ((2 x - 5) (2 x - 5))$

Step 1.1.2

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$5 x (2 x (2 x - 5) - 5 (2 x - 5))$

Step 1.1.2.2

Apply the distributive property.

$5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5))$

Step 1.1.2.3

Apply the distributive property.

$5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Rewrite using the commutative property of multiplication.

$5 x (2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Step 1.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.3.1.2.1

Move $x$ .

$5 x (2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Step 1.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.1.3.1.4

Multiply $- 5$ by $2$ .

$5 x (4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5)$

Step 1.1.3.1.5

Multiply $2$ by $- 5$ .

$5 x (4 x^{2} - 10 x - 10 x - 5 \cdot - 5)$

Step 1.1.3.1.6

Multiply $- 5$ by $- 5$ .

$5 x (4 x^{2} - 10 x - 10 x + 25)$

Step 1.1.3.2

Subtract $10 x$ from $- 10 x$ .

$5 x (4 x^{2} - 20 x + 25)$

Step 1.1.4

Apply the distributive property.

$5 x (4 x^{2}) + 5 x (- 20 x) + 5 x \cdot 25$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Rewrite using the commutative property of multiplication.

$5 \cdot 4 x \cdot x^{2} + 5 x (- 20 x) + 5 x \cdot 25$

Step 1.1.5.2

Rewrite using the commutative property of multiplication.

$5 \cdot 4 x \cdot x^{2} + 5 \cdot - 20 x \cdot x + 5 x \cdot 25$

Step 1.1.5.3

Multiply $25$ by $5$ .

$5 \cdot 4 x \cdot x^{2} + 5 \cdot - 20 x \cdot x + 125 x$

Step 1.1.6

Simplify each term.

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Step 1.1.6.1

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

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Step 1.1.6.1.1

Move $x^{2}$ .

$5 \cdot 4 (x^{2} x) + 5 \cdot - 20 x \cdot x + 125 x$

Step 1.1.6.1.2

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

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Step 1.1.6.1.2.1

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

$5 \cdot 4 (x^{2} x^{1}) + 5 \cdot - 20 x \cdot x + 125 x$

Step 1.1.6.1.2.2

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

$5 \cdot 4 x^{2 + 1} + 5 \cdot - 20 x \cdot x + 125 x$

Step 1.1.6.1.3

Add $2$ and $1$ .

$5 \cdot 4 x^{3} + 5 \cdot - 20 x \cdot x + 125 x$

Step 1.1.6.2

Multiply $5$ by $4$ .

$20 x^{3} + 5 \cdot - 20 x \cdot x + 125 x$

Step 1.1.6.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.6.3.1

Move $x$ .

$20 x^{3} + 5 \cdot - 20 (x \cdot x) + 125 x$

Step 1.1.6.3.2

Multiply $x$ by $x$ .

$20 x^{3} + 5 \cdot - 20 x^{2} + 125 x$

Step 1.1.6.4

Multiply $5$ by $- 20$ .

$20 x^{3} - 100 x^{2} + 125 x$

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

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$5 x ((2 x - 5) (2 x - 5))$

Step 3.1.2

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$5 x (2 x (2 x - 5) - 5 (2 x - 5))$

Step 3.1.2.2

Apply the distributive property.

$5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5))$

Step 3.1.2.3

Apply the distributive property.

$5 x (2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Rewrite using the commutative property of multiplication.

$5 x (2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Step 3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.3.1.2.1

Move $x$ .

$5 x (2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5)$

Step 3.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3.1.3

Multiply $2$ by $2$ .

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

Step 3.1.3.1.4

Multiply $- 5$ by $2$ .

$5 x (4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5)$

Step 3.1.3.1.5

Multiply $2$ by $- 5$ .

$5 x (4 x^{2} - 10 x - 10 x - 5 \cdot - 5)$

Step 3.1.3.1.6

Multiply $- 5$ by $- 5$ .

$5 x (4 x^{2} - 10 x - 10 x + 25)$

Step 3.1.3.2

Subtract $10 x$ from $- 10 x$ .

$5 x (4 x^{2} - 20 x + 25)$

Step 3.1.4

Apply the distributive property.

$5 x (4 x^{2}) + 5 x (- 20 x) + 5 x \cdot 25$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Rewrite using the commutative property of multiplication.

$5 \cdot 4 x \cdot x^{2} + 5 x (- 20 x) + 5 x \cdot 25$

Step 3.1.5.2

Rewrite using the commutative property of multiplication.

$5 \cdot 4 x \cdot x^{2} + 5 \cdot - 20 x \cdot x + 5 x \cdot 25$

Step 3.1.5.3

Multiply $25$ by $5$ .

$5 \cdot 4 x \cdot x^{2} + 5 \cdot - 20 x \cdot x + 125 x$

Step 3.1.6

Simplify each term.

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Step 3.1.6.1

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

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Step 3.1.6.1.1

Move $x^{2}$ .

$5 \cdot 4 (x^{2} x) + 5 \cdot - 20 x \cdot x + 125 x$

Step 3.1.6.1.2

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

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Step 3.1.6.1.2.1

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

$5 \cdot 4 (x^{2} x^{1}) + 5 \cdot - 20 x \cdot x + 125 x$

Step 3.1.6.1.2.2

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

$5 \cdot 4 x^{2 + 1} + 5 \cdot - 20 x \cdot x + 125 x$

Step 3.1.6.1.3

Add $2$ and $1$ .

$5 \cdot 4 x^{3} + 5 \cdot - 20 x \cdot x + 125 x$

Step 3.1.6.2

Multiply $5$ by $4$ .

$20 x^{3} + 5 \cdot - 20 x \cdot x + 125 x$

Step 3.1.6.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.6.3.1

Move $x$ .

$20 x^{3} + 5 \cdot - 20 (x \cdot x) + 125 x$

Step 3.1.6.3.2

Multiply $x$ by $x$ .

$20 x^{3} + 5 \cdot - 20 x^{2} + 125 x$

Step 3.1.6.4

Multiply $5$ by $- 20$ .

$20 x^{3} - 100 x^{2} + 125 x$

Step 3.2

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

$20 x^{3}$

Step 3.3

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

$20$

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