Find the Degree, Leading Term, and Leading Coefficient y=(x+4)(x+1)(x-1)^2

$y = (x + 4) (x + 1) {(x - 1)}^{2}$

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

Expand $(x + 4) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$y = (x (x + 1) + 4 (x + 1)) {(x - 1)}^{2}$

Apply the distributive property.

$y = (x \cdot x + x \cdot 1 + 4 (x + 1)) {(x - 1)}^{2}$

Apply the distributive property.

$y = (x \cdot x + x \cdot 1 + 4 x + 4 \cdot 1) {(x - 1)}^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + x \cdot 1 + 4 x + 4 \cdot 1) {(x - 1)}^{2}$

Multiply $x$ by $1$ .

$y = (x^{2} + x + 4 x + 4 \cdot 1) {(x - 1)}^{2}$

Multiply $4$ by $1$ .

$y = (x^{2} + x + 4 x + 4) {(x - 1)}^{2}$

Add $x$ and $4 x$ .

$y = (x^{2} + 5 x + 4) {(x - 1)}^{2}$

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$y = (x^{2} + 5 x + 4) ((x - 1) (x - 1))$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$y = (x^{2} + 5 x + 4) (x (x - 1) - 1 (x - 1))$

Apply the distributive property.

$y = (x^{2} + 5 x + 4) (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Apply the distributive property.

$y = (x^{2} + 5 x + 4) (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = (x^{2} + 5 x + 4) (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Move $- 1$ to the left of $x$ .

$y = (x^{2} + 5 x + 4) (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$y = (x^{2} + 5 x + 4) (x^{2} - x - 1 x - 1 \cdot - 1)$

Rewrite $- 1 x$ as $- x$ .

$y = (x^{2} + 5 x + 4) (x^{2} - x - x - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

$y = (x^{2} + 5 x + 4) (x^{2} - x - x + 1)$

Subtract $x$ from $- x$ .

$y = (x^{2} + 5 x + 4) (x^{2} - 2 x + 1)$

Expand $(x^{2} + 5 x + 4) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x^{2} + x^{2} (- 2 x) + x^{2} \cdot 1 + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Simplify terms.

Simplify each term.

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

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

$y = x^{2 + 2} + x^{2} (- 2 x) + x^{2} \cdot 1 + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Add $2$ and $2$ .

$y = x^{4} + x^{2} (- 2 x) + x^{2} \cdot 1 + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Rewrite using the commutative property of multiplication.

$y = x^{4} - 2 x^{2} x + x^{2} \cdot 1 + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

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

Move $x$ .

$y = x^{4} - 2 (x \cdot x^{2}) + x^{2} \cdot 1 + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

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

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

$y = x^{4} - 2 (x \cdot x^{2}) + x^{2} \cdot 1 + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

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

$y = x^{4} - 2 x^{1 + 2} + x^{2} \cdot 1 + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Add $1$ and $2$ .

$y = x^{4} - 2 x^{3} + x^{2} \cdot 1 + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Multiply $x^{2}$ by $1$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x \cdot x^{2} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

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

Move $x^{2}$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 (x^{2} x) + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

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

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

$y = x^{4} - 2 x^{3} + x^{2} + 5 (x^{2} x) + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

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

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{2 + 1} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Add $2$ and $1$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} + 5 x (- 2 x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Rewrite using the commutative property of multiplication.

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} + 5 \cdot (- 2 x \cdot x) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

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

Move $x$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} + 5 \cdot (- 2 (x \cdot x)) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Multiply $x$ by $x$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} + 5 \cdot (- 2 x^{2}) + 5 x \cdot 1 + 4 x^{2} + 4 (- 2 x) + 4 \cdot 1$

Multiply $5$ by $- 2$ .

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

Multiply $5$ by $1$ .

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

Multiply $- 2$ by $4$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} - 10 x^{2} + 5 x + 4 x^{2} - 8 x + 4 \cdot 1$

Multiply $4$ by $1$ .

$y = x^{4} - 2 x^{3} + x^{2} + 5 x^{3} - 10 x^{2} + 5 x + 4 x^{2} - 8 x + 4$

Simplify by adding terms.

Add $- 2 x^{3}$ and $5 x^{3}$ .

$y = x^{4} + 3 x^{3} + x^{2} - 10 x^{2} + 5 x + 4 x^{2} - 8 x + 4$

Subtract $10 x^{2}$ from $x^{2}$ .

$y = x^{4} + 3 x^{3} - 9 x^{2} + 5 x + 4 x^{2} - 8 x + 4$

Add $- 9 x^{2}$ and $4 x^{2}$ .

$y = x^{4} + 3 x^{3} - 5 x^{2} + 5 x - 8 x + 4$

Subtract $8 x$ from $5 x$ .

$y = x^{4} + 3 x^{3} - 5 x^{2} - 3 x + 4$

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

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

$x^{4} \to 4$

$3 x^{3} \to 3$

$- 5 x^{2} \to 2$

$- 3 x \to 1$

$4 \to 0$

The largest exponent is the degree of the polynomial.

$4$

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

$x^{4}$

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

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

$x^{4}$

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

$1$

List the results.

Polynomial Degree: $4$

Leading Term: $x^{4}$

Leading Coefficient: $1$

Find the Degree, Leading Term, and Leading Coefficient y=(x+4)(x+1)(x-1)^2

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