Algebra Examples

$6 {(x - 1)}^{11} {(x + 1)}^{4}$

Step 1

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

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Step 1.1

Use the Binomial Theorem.

$6 {(x - 1)}^{11} (x^{4} + 4 x^{3} \cdot 1 + 6 x^{2} \cdot 1^{2} + 4 x \cdot 1^{3} + 1^{4})$

Step 1.2

Simplify terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $4$ by $1$ .

$6 {(x - 1)}^{11} (x^{4} + 4 x^{3} + 6 x^{2} \cdot 1^{2} + 4 x \cdot 1^{3} + 1^{4})$

Step 1.2.1.2

One to any power is one.

$6 {(x - 1)}^{11} (x^{4} + 4 x^{3} + 6 x^{2} \cdot 1 + 4 x \cdot 1^{3} + 1^{4})$

Step 1.2.1.3

Multiply $6$ by $1$ .

$6 {(x - 1)}^{11} (x^{4} + 4 x^{3} + 6 x^{2} + 4 x \cdot 1^{3} + 1^{4})$

Step 1.2.1.4

One to any power is one.

$6 {(x - 1)}^{11} (x^{4} + 4 x^{3} + 6 x^{2} + 4 x \cdot 1 + 1^{4})$

Step 1.2.1.5

Multiply $4$ by $1$ .

$6 {(x - 1)}^{11} (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1^{4})$

Step 1.2.1.6

One to any power is one.

$6 {(x - 1)}^{11} (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1)$

Step 1.2.2

Apply the distributive property.

$6 {(x - 1)}^{11} x^{4} + 6 {(x - 1)}^{11} (4 x^{3}) + 6 {(x - 1)}^{11} (6 x^{2}) + 6 {(x - 1)}^{11} (4 x) + 6 {(x - 1)}^{11} \cdot 1$

Step 1.3

Simplify.

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Step 1.3.1

Rewrite using the commutative property of multiplication.

$6 {(x - 1)}^{11} x^{4} + 6 \cdot 4 {(x - 1)}^{11} x^{3} + 6 {(x - 1)}^{11} (6 x^{2}) + 6 {(x - 1)}^{11} (4 x) + 6 {(x - 1)}^{11} \cdot 1$

Step 1.3.2

Rewrite using the commutative property of multiplication.

$6 {(x - 1)}^{11} x^{4} + 6 \cdot 4 {(x - 1)}^{11} x^{3} + 6 \cdot 6 {(x - 1)}^{11} x^{2} + 6 {(x - 1)}^{11} (4 x) + 6 {(x - 1)}^{11} \cdot 1$

Step 1.3.3

Rewrite using the commutative property of multiplication.

$6 {(x - 1)}^{11} x^{4} + 6 \cdot 4 {(x - 1)}^{11} x^{3} + 6 \cdot 6 {(x - 1)}^{11} x^{2} + 6 \cdot 4 {(x - 1)}^{11} x + 6 {(x - 1)}^{11} \cdot 1$

Step 1.3.4

Multiply $6$ by $1$ .

$6 {(x - 1)}^{11} x^{4} + 6 \cdot 4 {(x - 1)}^{11} x^{3} + 6 \cdot 6 {(x - 1)}^{11} x^{2} + 6 \cdot 4 {(x - 1)}^{11} x + 6 {(x - 1)}^{11}$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $6$ by $4$ .

$6 {(x - 1)}^{11} x^{4} + 24 {(x - 1)}^{11} x^{3} + 6 \cdot 6 {(x - 1)}^{11} x^{2} + 6 \cdot 4 {(x - 1)}^{11} x + 6 {(x - 1)}^{11}$

Step 1.4.2

Multiply $6$ by $6$ .

$6 {(x - 1)}^{11} x^{4} + 24 {(x - 1)}^{11} x^{3} + 36 {(x - 1)}^{11} x^{2} + 6 \cdot 4 {(x - 1)}^{11} x + 6 {(x - 1)}^{11}$

Step 1.4.3

Multiply $6$ by $4$ .

$6 {(x - 1)}^{11} x^{4} + 24 {(x - 1)}^{11} x^{3} + 36 {(x - 1)}^{11} x^{2} + 24 {(x - 1)}^{11} x + 6 {(x - 1)}^{11}$

Step 1.5

Reorder factors in $6 {(x - 1)}^{11} x^{4} + 24 {(x - 1)}^{11} x^{3} + 36 {(x - 1)}^{11} x^{2} + 24 {(x - 1)}^{11} x + 6 {(x - 1)}^{11}$ .

$6 x^{4} {(x - 1)}^{11} + 24 x^{3} {(x - 1)}^{11} + 36 x^{2} {(x - 1)}^{11} + 24 x {(x - 1)}^{11} + 6 {(x - 1)}^{11}$

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.

$6 x^{4} {(x - 1)}^{11} \to 15$

$24 x^{3} {(x - 1)}^{11} \to 14$

$36 x^{2} {(x - 1)}^{11} \to 13$

$24 x {(x - 1)}^{11} \to 12$

$6 {(x - 1)}^{11} \to 11$

Step 2.2

The largest exponent is the degree of the polynomial.

$15$

Step 3

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

$6 x^{4} {(x - 1)}^{11}$

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.

$6 x^{4} {(x - 1)}^{11}$

Step 4.2

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

$6$

Step 5

List the results.

Polynomial Degree: $15$

Leading Term: $6 x^{4} {(x - 1)}^{11}$

Leading Coefficient: $6$