Write in Standard Form G(x)=4(x-3)^2(x^2+1)

$G (x) = 4 {(x - 3)}^{2} (x^{2} + 1)$

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 ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$G (x) = 4 ((x - 3) (x - 3)) (x^{2} + 1)$

Step 3

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$G (x) = 4 (x (x - 3) - 3 (x - 3)) (x^{2} + 1)$

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $- 3$ by $- 3$ .

$G (x) = 4 (x^{2} - 3 x - 3 x + 9) (x^{2} + 1)$

Subtract $3 x$ from $- 3 x$ .

$G (x) = 4 (x^{2} - 6 x + 9) (x^{2} + 1)$

Step 5

Apply the distributive property.

$G (x) = (4 x^{2} + 4 (- 6 x) + 4 \cdot 9) (x^{2} + 1)$

Step 6

Simplify.

Multiply $- 6$ by $4$ .

$G (x) = (4 x^{2} - 24 x + 4 \cdot 9) (x^{2} + 1)$

Multiply $4$ by $9$ .

$G (x) = (4 x^{2} - 24 x + 36) (x^{2} + 1)$

Step 7

Expand $(4 x^{2} - 24 x + 36) (x^{2} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$G (x) = 4 x^{2} x^{2} + 4 x^{2} \cdot 1 - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Step 8

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$G (x) = 4 (x^{2} x^{2}) + 4 x^{2} \cdot 1 - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

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

$G (x) = 4 x^{2 + 2} + 4 x^{2} \cdot 1 - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Add $2$ and $2$ .

$G (x) = 4 x^{4} + 4 x^{2} \cdot 1 - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Multiply $4$ by $1$ .

$G (x) = 4 x^{4} + 4 x^{2} - 24 x \cdot x^{2} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

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

Move $x^{2}$ .

$G (x) = 4 x^{4} + 4 x^{2} - 24 (x^{2} x) - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

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

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

$G (x) = 4 x^{4} + 4 x^{2} - 24 (x^{2} x) - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

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

$G (x) = 4 x^{4} + 4 x^{2} - 24 x^{2 + 1} - 24 x \cdot 1 + 36 x^{2} + 36 \cdot 1$

Add $2$ and $1$ .

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

Multiply $- 24$ by $1$ .

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

Multiply $36$ by $1$ .

$G (x) = 4 x^{4} + 4 x^{2} - 24 x^{3} - 24 x + 36 x^{2} + 36$

Simplify by adding terms.

Add $4 x^{2}$ and $36 x^{2}$ .

$G (x) = 4 x^{4} + 40 x^{2} - 24 x^{3} - 24 x + 36$

Move $40 x^{2}$ .

$G (x) = 4 x^{4} - 24 x^{3} + 40 x^{2} - 24 x + 36$

Write in Standard Form G(x)=4(x-3)^2(x^2+1)

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