Algebra Examples

$3 n^{2} (n^{2} + 4 n - 5) - (2 n^{2} - n^{4} + 3)$

Step 1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$3 n^{2} n^{2} + 3 n^{2} (4 n) + 3 n^{2} \cdot - 5 - (2 n^{2} - n^{4} + 3)$

Step 1.1.2

Simplify.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

Move $n^{2}$ .

$3 (n^{2} n^{2}) + 3 n^{2} (4 n) + 3 n^{2} \cdot - 5 - (2 n^{2} - n^{4} + 3)$

Step 1.1.2.1.2

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

$3 n^{2 + 2} + 3 n^{2} (4 n) + 3 n^{2} \cdot - 5 - (2 n^{2} - n^{4} + 3)$

Step 1.1.2.1.3

Add $2$ and $2$ .

$3 n^{4} + 3 n^{2} (4 n) + 3 n^{2} \cdot - 5 - (2 n^{2} - n^{4} + 3)$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$3 n^{4} + 3 \cdot 4 n^{2} n + 3 n^{2} \cdot - 5 - (2 n^{2} - n^{4} + 3)$

Step 1.1.2.3

Multiply $- 5$ by $3$ .

$3 n^{4} + 3 \cdot 4 n^{2} n - 15 n^{2} - (2 n^{2} - n^{4} + 3)$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

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

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Step 1.1.3.1.1

Move $n$ .

$3 n^{4} + 3 \cdot 4 (n \cdot n^{2}) - 15 n^{2} - (2 n^{2} - n^{4} + 3)$

Step 1.1.3.1.2

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

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Step 1.1.3.1.2.1

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

$3 n^{4} + 3 \cdot 4 (n^{1} n^{2}) - 15 n^{2} - (2 n^{2} - n^{4} + 3)$

Step 1.1.3.1.2.2

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

$3 n^{4} + 3 \cdot 4 n^{1 + 2} - 15 n^{2} - (2 n^{2} - n^{4} + 3)$

Step 1.1.3.1.3

Add $1$ and $2$ .

$3 n^{4} + 3 \cdot 4 n^{3} - 15 n^{2} - (2 n^{2} - n^{4} + 3)$

Step 1.1.3.2

Multiply $3$ by $4$ .

$3 n^{4} + 12 n^{3} - 15 n^{2} - (2 n^{2} - n^{4} + 3)$

Step 1.1.4

Apply the distributive property.

$3 n^{4} + 12 n^{3} - 15 n^{2} - (2 n^{2}) - - n^{4} - 1 \cdot 3$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply $2$ by $- 1$ .

$3 n^{4} + 12 n^{3} - 15 n^{2} - 2 n^{2} - - n^{4} - 1 \cdot 3$

Step 1.1.5.2

Multiply $- - n^{4}$ .

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Step 1.1.5.2.1

Multiply $- 1$ by $- 1$ .

$3 n^{4} + 12 n^{3} - 15 n^{2} - 2 n^{2} + 1 n^{4} - 1 \cdot 3$

Step 1.1.5.2.2

Multiply $n^{4}$ by $1$ .

$3 n^{4} + 12 n^{3} - 15 n^{2} - 2 n^{2} + n^{4} - 1 \cdot 3$

Step 1.1.5.3

Multiply $- 1$ by $3$ .

$3 n^{4} + 12 n^{3} - 15 n^{2} - 2 n^{2} + n^{4} - 3$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Add $3 n^{4}$ and $n^{4}$ .

$4 n^{4} + 12 n^{3} - 15 n^{2} - 2 n^{2} - 3$

Step 1.2.2

Subtract $2 n^{2}$ from $- 15 n^{2}$ .

$4 n^{4} + 12 n^{3} - 17 n^{2} - 3$

Step 2

The degree of a polynomial is the highest degree of its terms.

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Step 2.1

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$4 n^{4} \to 4$

$12 n^{3} \to 3$

$- 17 n^{2} \to 2$

$- 3 \to 0$

Step 2.2

The largest exponent is the degree of the polynomial.

$4$

Step 3

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

$4 n^{4}$

Step 4

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

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Step 4.1

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

$4 n^{4}$

Step 4.2

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

$4$

Step 5

List the results.

Polynomial Degree: $4$

Leading Term: $4 n^{4}$

Leading Coefficient: $4$