Algebra Examples

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

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

Simplify terms.

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Simplify the expression.

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Step 1.1.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.1.2.2

Multiply $2$ by $5$ .

$(5 \cdot 3 x^{2} x + 10 x^{2}) (x - 5)$

Step 1.1.1.3

Simplify each term.

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Step 1.1.1.3.1

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

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Step 1.1.1.3.1.1

Move $x$ .

$(5 \cdot 3 (x \cdot x^{2}) + 10 x^{2}) (x - 5)$

Step 1.1.1.3.1.2

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

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Step 1.1.1.3.1.2.1

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

$(5 \cdot 3 (x^{1} x^{2}) + 10 x^{2}) (x - 5)$

Step 1.1.1.3.1.2.2

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

$(5 \cdot 3 x^{1 + 2} + 10 x^{2}) (x - 5)$

Step 1.1.1.3.1.3

Add $1$ and $2$ .

$(5 \cdot 3 x^{3} + 10 x^{2}) (x - 5)$

Step 1.1.1.3.2

Multiply $5$ by $3$ .

$(15 x^{3} + 10 x^{2}) (x - 5)$

Step 1.1.2

Expand $(15 x^{3} + 10 x^{2}) (x - 5)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$15 x^{3} (x - 5) + 10 x^{2} (x - 5)$

Step 1.1.2.2

Apply the distributive property.

$15 x^{3} x + 15 x^{3} \cdot - 5 + 10 x^{2} (x - 5)$

Step 1.1.2.3

Apply the distributive property.

$15 x^{3} x + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

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 $x^{3}$ by $x$ by adding the exponents.

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Step 1.1.3.1.1.1

Move $x$ .

$15 (x \cdot x^{3}) + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 1.1.3.1.1.2

Multiply $x$ by $x^{3}$ .

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Step 1.1.3.1.1.2.1

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

$15 (x^{1} x^{3}) + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 1.1.3.1.1.2.2

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

$15 x^{1 + 3} + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 1.1.3.1.1.3

Add $1$ and $3$ .

$15 x^{4} + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 1.1.3.1.2

Multiply $- 5$ by $15$ .

$15 x^{4} - 75 x^{3} + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 1.1.3.1.3

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

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Step 1.1.3.1.3.1

Move $x$ .

$15 x^{4} - 75 x^{3} + 10 (x \cdot x^{2}) + 10 x^{2} \cdot - 5$

Step 1.1.3.1.3.2

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

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Step 1.1.3.1.3.2.1

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

$15 x^{4} - 75 x^{3} + 10 (x^{1} x^{2}) + 10 x^{2} \cdot - 5$

Step 1.1.3.1.3.2.2

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

$15 x^{4} - 75 x^{3} + 10 x^{1 + 2} + 10 x^{2} \cdot - 5$

Step 1.1.3.1.3.3

Add $1$ and $2$ .

$15 x^{4} - 75 x^{3} + 10 x^{3} + 10 x^{2} \cdot - 5$

Step 1.1.3.1.4

Multiply $- 5$ by $10$ .

$15 x^{4} - 75 x^{3} + 10 x^{3} - 50 x^{2}$

Step 1.1.3.2

Add $- 75 x^{3}$ and $10 x^{3}$ .

$15 x^{4} - 65 x^{3} - 50 x^{2}$

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

Simplify terms.

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Step 3.1.1.1

Apply the distributive property.

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

Step 3.1.1.2

Simplify the expression.

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Step 3.1.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.1.2.2

Multiply $2$ by $5$ .

$(5 \cdot 3 x^{2} x + 10 x^{2}) (x - 5)$

Step 3.1.1.3

Simplify each term.

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Step 3.1.1.3.1

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

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Step 3.1.1.3.1.1

Move $x$ .

$(5 \cdot 3 (x \cdot x^{2}) + 10 x^{2}) (x - 5)$

Step 3.1.1.3.1.2

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

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Step 3.1.1.3.1.2.1

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

$(5 \cdot 3 (x^{1} x^{2}) + 10 x^{2}) (x - 5)$

Step 3.1.1.3.1.2.2

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

$(5 \cdot 3 x^{1 + 2} + 10 x^{2}) (x - 5)$

Step 3.1.1.3.1.3

Add $1$ and $2$ .

$(5 \cdot 3 x^{3} + 10 x^{2}) (x - 5)$

Step 3.1.1.3.2

Multiply $5$ by $3$ .

$(15 x^{3} + 10 x^{2}) (x - 5)$

Step 3.1.2

Expand $(15 x^{3} + 10 x^{2}) (x - 5)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$15 x^{3} (x - 5) + 10 x^{2} (x - 5)$

Step 3.1.2.2

Apply the distributive property.

$15 x^{3} x + 15 x^{3} \cdot - 5 + 10 x^{2} (x - 5)$

Step 3.1.2.3

Apply the distributive property.

$15 x^{3} x + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

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 $x^{3}$ by $x$ by adding the exponents.

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Step 3.1.3.1.1.1

Move $x$ .

$15 (x \cdot x^{3}) + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 3.1.3.1.1.2

Multiply $x$ by $x^{3}$ .

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Step 3.1.3.1.1.2.1

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

$15 (x^{1} x^{3}) + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 3.1.3.1.1.2.2

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

$15 x^{1 + 3} + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 3.1.3.1.1.3

Add $1$ and $3$ .

$15 x^{4} + 15 x^{3} \cdot - 5 + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 3.1.3.1.2

Multiply $- 5$ by $15$ .

$15 x^{4} - 75 x^{3} + 10 x^{2} x + 10 x^{2} \cdot - 5$

Step 3.1.3.1.3

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

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Step 3.1.3.1.3.1

Move $x$ .

$15 x^{4} - 75 x^{3} + 10 (x \cdot x^{2}) + 10 x^{2} \cdot - 5$

Step 3.1.3.1.3.2

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

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Step 3.1.3.1.3.2.1

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

$15 x^{4} - 75 x^{3} + 10 (x^{1} x^{2}) + 10 x^{2} \cdot - 5$

Step 3.1.3.1.3.2.2

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

$15 x^{4} - 75 x^{3} + 10 x^{1 + 2} + 10 x^{2} \cdot - 5$

Step 3.1.3.1.3.3

Add $1$ and $2$ .

$15 x^{4} - 75 x^{3} + 10 x^{3} + 10 x^{2} \cdot - 5$

Step 3.1.3.1.4

Multiply $- 5$ by $10$ .

$15 x^{4} - 75 x^{3} + 10 x^{3} - 50 x^{2}$

Step 3.1.3.2

Add $- 75 x^{3}$ and $10 x^{3}$ .

$15 x^{4} - 65 x^{3} - 50 x^{2}$

Step 3.2

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

$15 x^{4}$

Step 3.3

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

$15$

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