Find the Degree, Leading Term, and Leading Coefficient g(x)=4(x-1)^2(x^2+5)

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

Step 1

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $- 2$ by $4$ .

$g (x) = (4 x^{2} - 8 x + 4 \cdot 1) (x^{2} + 5)$

Multiply $4$ by $1$ .

$g (x) = (4 x^{2} - 8 x + 4) (x^{2} + 5)$

Expand $(4 x^{2} - 8 x + 4) (x^{2} + 5)$ by multiplying each term in the first expression by each term in the second expression.

$g (x) = 4 x^{2} x^{2} + 4 x^{2} \cdot 5 - 8 x \cdot x^{2} - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$g (x) = 4 (x^{2} x^{2}) + 4 x^{2} \cdot 5 - 8 x \cdot x^{2} - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

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

$g (x) = 4 x^{2 + 2} + 4 x^{2} \cdot 5 - 8 x \cdot x^{2} - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

Add $2$ and $2$ .

$g (x) = 4 x^{4} + 4 x^{2} \cdot 5 - 8 x \cdot x^{2} - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

Multiply $5$ by $4$ .

$g (x) = 4 x^{4} + 20 x^{2} - 8 x \cdot x^{2} - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

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

Move $x^{2}$ .

$g (x) = 4 x^{4} + 20 x^{2} - 8 (x^{2} x) - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

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

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

$g (x) = 4 x^{4} + 20 x^{2} - 8 (x^{2} x) - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

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

$g (x) = 4 x^{4} + 20 x^{2} - 8 x^{2 + 1} - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

Add $2$ and $1$ .

$g (x) = 4 x^{4} + 20 x^{2} - 8 x^{3} - 8 x \cdot 5 + 4 x^{2} + 4 \cdot 5$

Multiply $5$ by $- 8$ .

$g (x) = 4 x^{4} + 20 x^{2} - 8 x^{3} - 40 x + 4 x^{2} + 4 \cdot 5$

Multiply $4$ by $5$ .

$g (x) = 4 x^{4} + 20 x^{2} - 8 x^{3} - 40 x + 4 x^{2} + 20$

Simplify by adding terms.

Add $20 x^{2}$ and $4 x^{2}$ .

$g (x) = 4 x^{4} + 24 x^{2} - 8 x^{3} - 40 x + 20$

Move $24 x^{2}$ .

$g (x) = 4 x^{4} - 8 x^{3} + 24 x^{2} - 40 x + 20$

Step 2

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.

$4 x^{4} \to 4$

$- 8 x^{3} \to 3$

$24 x^{2} \to 2$

$- 40 x \to 1$

$20 \to 0$

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.

$4 x^{4}$

Step 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.

$4 x^{4}$

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

$4$

Step 5

List the results.

Polynomial Degree: $4$

Leading Term: $4 x^{4}$

Leading Coefficient: $4$

Find the Degree, Leading Term, and Leading Coefficient g(x)=4(x-1)^2(x^2+5)

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