Write in Standard Form g(x)=2(x-1)^2(x^2+5)

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

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$g (x) = a x^{2} + b x + c$

Step 2

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

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

Step 3

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- x$ .

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

Step 5

Apply the distributive property.

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

Step 6

Simplify.

Multiply $- 2$ by $2$ .

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

Multiply $2$ by $1$ .

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

Step 7

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

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

Step 8

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $5$ by $2$ .

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $5$ by $- 4$ .

$g (x) = 2 x^{4} + 10 x^{2} - 4 x^{3} - 20 x + 2 x^{2} + 2 \cdot 5$

Multiply $2$ by $5$ .

$g (x) = 2 x^{4} + 10 x^{2} - 4 x^{3} - 20 x + 2 x^{2} + 10$

Simplify by adding terms.

Add $10 x^{2}$ and $2 x^{2}$ .

$g (x) = 2 x^{4} + 12 x^{2} - 4 x^{3} - 20 x + 10$

Move $12 x^{2}$ .

$g (x) = 2 x^{4} - 4 x^{3} + 12 x^{2} - 20 x + 10$

Write in Standard Form g(x)=2(x-1)^2(x^2+5)

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