Algebra Examples

$f (x) = 3 (x - 1) {(x + 5)}^{2}$

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

Simplify by multiplying through.

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Step 1.1.1.1

Apply the distributive property.

$(3 x + 3 \cdot - 1) {(x + 5)}^{2}$

Step 1.1.1.2

Simplify the expression.

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Step 1.1.1.2.1

Multiply $3$ by $- 1$ .

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

Step 1.1.1.2.2

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

$(3 x - 3) ((x + 5) (x + 5))$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$(3 x - 3) (x (x + 5) + 5 (x + 5))$

Step 1.1.2.2

Apply the distributive property.

$(3 x - 3) (x \cdot x + x \cdot 5 + 5 (x + 5))$

Step 1.1.2.3

Apply the distributive property.

$(3 x - 3) (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5)$

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

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

Multiply $5$ by $5$ .

$(3 x - 3) (x^{2} + 5 x + 5 x + 25)$

Step 1.1.3.2

Add $5 x$ and $5 x$ .

$(3 x - 3) (x^{2} + 10 x + 25)$

Step 1.1.4

Expand $(3 x - 3) (x^{2} + 10 x + 25)$ by multiplying each term in the first expression by each term in the second expression.

$3 x \cdot x^{2} + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5

Simplify terms.

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Step 1.1.5.1

Simplify each term.

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Step 1.1.5.1.1

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

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Step 1.1.5.1.1.1

Move $x^{2}$ .

$3 (x^{2} x) + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.1.2

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

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Step 1.1.5.1.1.2.1

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

$3 (x^{2} x^{1}) + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.1.2.2

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

$3 x^{2 + 1} + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.1.3

Add $2$ and $1$ .

$3 x^{3} + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.2

Rewrite using the commutative property of multiplication.

$3 x^{3} + 3 \cdot 10 x \cdot x + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.3

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

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Step 1.1.5.1.3.1

Move $x$ .

$3 x^{3} + 3 \cdot 10 (x \cdot x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.3.2

Multiply $x$ by $x$ .

$3 x^{3} + 3 \cdot 10 x^{2} + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.4

Multiply $3$ by $10$ .

$3 x^{3} + 30 x^{2} + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.5

Multiply $25$ by $3$ .

$3 x^{3} + 30 x^{2} + 75 x - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 1.1.5.1.6

Multiply $10$ by $- 3$ .

$3 x^{3} + 30 x^{2} + 75 x - 3 x^{2} - 30 x - 3 \cdot 25$

Step 1.1.5.1.7

Multiply $- 3$ by $25$ .

$3 x^{3} + 30 x^{2} + 75 x - 3 x^{2} - 30 x - 75$

Step 1.1.5.2

Simplify by adding terms.

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Step 1.1.5.2.1

Subtract $3 x^{2}$ from $30 x^{2}$ .

$3 x^{3} + 27 x^{2} + 75 x - 30 x - 75$

Step 1.1.5.2.2

Subtract $30 x$ from $75 x$ .

$3 x^{3} + 27 x^{2} + 45 x - 75$

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

Simplify by multiplying through.

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Step 3.1.1.1

Apply the distributive property.

$(3 x + 3 \cdot - 1) {(x + 5)}^{2}$

Step 3.1.1.2

Simplify the expression.

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Step 3.1.1.2.1

Multiply $3$ by $- 1$ .

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

Step 3.1.1.2.2

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

$(3 x - 3) ((x + 5) (x + 5))$

Step 3.1.2

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

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Step 3.1.2.1

Apply the distributive property.

$(3 x - 3) (x (x + 5) + 5 (x + 5))$

Step 3.1.2.2

Apply the distributive property.

$(3 x - 3) (x \cdot x + x \cdot 5 + 5 (x + 5))$

Step 3.1.2.3

Apply the distributive property.

$(3 x - 3) (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5)$

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

Multiply $x$ by $x$ .

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

Step 3.1.3.1.2

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

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

Step 3.1.3.1.3

Multiply $5$ by $5$ .

$(3 x - 3) (x^{2} + 5 x + 5 x + 25)$

Step 3.1.3.2

Add $5 x$ and $5 x$ .

$(3 x - 3) (x^{2} + 10 x + 25)$

Step 3.1.4

Expand $(3 x - 3) (x^{2} + 10 x + 25)$ by multiplying each term in the first expression by each term in the second expression.

$3 x \cdot x^{2} + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5

Simplify terms.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

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

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Step 3.1.5.1.1.1

Move $x^{2}$ .

$3 (x^{2} x) + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.1.2

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

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Step 3.1.5.1.1.2.1

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

$3 (x^{2} x^{1}) + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.1.2.2

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

$3 x^{2 + 1} + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.1.3

Add $2$ and $1$ .

$3 x^{3} + 3 x (10 x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.2

Rewrite using the commutative property of multiplication.

$3 x^{3} + 3 \cdot 10 x \cdot x + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.3

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

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Step 3.1.5.1.3.1

Move $x$ .

$3 x^{3} + 3 \cdot 10 (x \cdot x) + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.3.2

Multiply $x$ by $x$ .

$3 x^{3} + 3 \cdot 10 x^{2} + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.4

Multiply $3$ by $10$ .

$3 x^{3} + 30 x^{2} + 3 x \cdot 25 - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.5

Multiply $25$ by $3$ .

$3 x^{3} + 30 x^{2} + 75 x - 3 x^{2} - 3 (10 x) - 3 \cdot 25$

Step 3.1.5.1.6

Multiply $10$ by $- 3$ .

$3 x^{3} + 30 x^{2} + 75 x - 3 x^{2} - 30 x - 3 \cdot 25$

Step 3.1.5.1.7

Multiply $- 3$ by $25$ .

$3 x^{3} + 30 x^{2} + 75 x - 3 x^{2} - 30 x - 75$

Step 3.1.5.2

Simplify by adding terms.

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Step 3.1.5.2.1

Subtract $3 x^{2}$ from $30 x^{2}$ .

$3 x^{3} + 27 x^{2} + 75 x - 30 x - 75$

Step 3.1.5.2.2

Subtract $30 x$ from $75 x$ .

$3 x^{3} + 27 x^{2} + 45 x - 75$

Step 3.2

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

$3 x^{3}$

Step 3.3

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

$3$

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