Algebra Examples

$f (x) = x^{2} {(3 x - 5)}^{2}$

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 ${(3 x - 5)}^{2}$ as $(3 x - 5) (3 x - 5)$ .

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

Step 1.1.2

Expand $(3 x - 5) (3 x - 5)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

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.

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

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 $x$ .

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

Step 1.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.1.3.1.3

Multiply $3$ by $3$ .

$x^{2} (9 x^{2} + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5)$

Step 1.1.3.1.4

Multiply $- 5$ by $3$ .

$x^{2} (9 x^{2} - 15 x - 5 (3 x) - 5 \cdot - 5)$

Step 1.1.3.1.5

Multiply $3$ by $- 5$ .

$x^{2} (9 x^{2} - 15 x - 15 x - 5 \cdot - 5)$

Step 1.1.3.1.6

Multiply $- 5$ by $- 5$ .

$x^{2} (9 x^{2} - 15 x - 15 x + 25)$

Step 1.1.3.2

Subtract $15 x$ from $- 15 x$ .

$x^{2} (9 x^{2} - 30 x + 25)$

Step 1.1.4

Apply the distributive property.

$x^{2} (9 x^{2}) + x^{2} (- 30 x) + x^{2} \cdot 25$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Rewrite using the commutative property of multiplication.

$9 x^{2} x^{2} + x^{2} (- 30 x) + x^{2} \cdot 25$

Step 1.1.5.2

Rewrite using the commutative property of multiplication.

$9 x^{2} x^{2} - 30 x^{2} x + x^{2} \cdot 25$

Step 1.1.5.3

Move $25$ to the left of $x^{2}$ .

$9 x^{2} x^{2} - 30 x^{2} x + 25 \cdot x^{2}$

Step 1.1.6

Simplify each term.

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Step 1.1.6.1

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

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Step 1.1.6.1.1

Move $x^{2}$ .

$9 (x^{2} x^{2}) - 30 x^{2} x + 25 \cdot x^{2}$

Step 1.1.6.1.2

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

$9 x^{2 + 2} - 30 x^{2} x + 25 \cdot x^{2}$

Step 1.1.6.1.3

Add $2$ and $2$ .

$9 x^{4} - 30 x^{2} x + 25 \cdot x^{2}$

Step 1.1.6.2

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

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Step 1.1.6.2.1

Move $x$ .

$9 x^{4} - 30 (x \cdot x^{2}) + 25 \cdot x^{2}$

Step 1.1.6.2.2

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

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Step 1.1.6.2.2.1

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

$9 x^{4} - 30 (x^{1} x^{2}) + 25 \cdot x^{2}$

Step 1.1.6.2.2.2

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

$9 x^{4} - 30 x^{1 + 2} + 25 \cdot x^{2}$

Step 1.1.6.2.3

Add $1$ and $2$ .

$9 x^{4} - 30 x^{3} + 25 \cdot x^{2}$

$9 x^{4} - 30 x^{3} + 25 x^{2}$

Step 1.2

The largest exponent is the degree of the polynomial.

$4$

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

Rewrite ${(3 x - 5)}^{2}$ as $(3 x - 5) (3 x - 5)$ .

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

Step 3.1.2

Expand $(3 x - 5) (3 x - 5)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

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

Step 3.1.2.2

Apply the distributive property.

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

Step 3.1.2.3

Apply the distributive property.

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

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.

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

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 $x$ .

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

Step 3.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.3.1.3

Multiply $3$ by $3$ .

$x^{2} (9 x^{2} + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5)$

Step 3.1.3.1.4

Multiply $- 5$ by $3$ .

$x^{2} (9 x^{2} - 15 x - 5 (3 x) - 5 \cdot - 5)$

Step 3.1.3.1.5

Multiply $3$ by $- 5$ .

$x^{2} (9 x^{2} - 15 x - 15 x - 5 \cdot - 5)$

Step 3.1.3.1.6

Multiply $- 5$ by $- 5$ .

$x^{2} (9 x^{2} - 15 x - 15 x + 25)$

Step 3.1.3.2

Subtract $15 x$ from $- 15 x$ .

$x^{2} (9 x^{2} - 30 x + 25)$

Step 3.1.4

Apply the distributive property.

$x^{2} (9 x^{2}) + x^{2} (- 30 x) + x^{2} \cdot 25$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Rewrite using the commutative property of multiplication.

$9 x^{2} x^{2} + x^{2} (- 30 x) + x^{2} \cdot 25$

Step 3.1.5.2

Rewrite using the commutative property of multiplication.

$9 x^{2} x^{2} - 30 x^{2} x + x^{2} \cdot 25$

Step 3.1.5.3

Move $25$ to the left of $x^{2}$ .

$9 x^{2} x^{2} - 30 x^{2} x + 25 \cdot x^{2}$

Step 3.1.6

Simplify each term.

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Step 3.1.6.1

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

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Step 3.1.6.1.1

Move $x^{2}$ .

$9 (x^{2} x^{2}) - 30 x^{2} x + 25 \cdot x^{2}$

Step 3.1.6.1.2

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

$9 x^{2 + 2} - 30 x^{2} x + 25 \cdot x^{2}$

Step 3.1.6.1.3

Add $2$ and $2$ .

$9 x^{4} - 30 x^{2} x + 25 \cdot x^{2}$

Step 3.1.6.2

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

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Step 3.1.6.2.1

Move $x$ .

$9 x^{4} - 30 (x \cdot x^{2}) + 25 \cdot x^{2}$

Step 3.1.6.2.2

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

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Step 3.1.6.2.2.1

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

$9 x^{4} - 30 (x^{1} x^{2}) + 25 \cdot x^{2}$

Step 3.1.6.2.2.2

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

$9 x^{4} - 30 x^{1 + 2} + 25 \cdot x^{2}$

Step 3.1.6.2.3

Add $1$ and $2$ .

$9 x^{4} - 30 x^{3} + 25 \cdot x^{2}$

$9 x^{4} - 30 x^{3} + 25 x^{2}$

Step 3.2

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

$9 x^{4}$

Step 3.3

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

$9$

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