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

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

Step 1

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

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

Step 2

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

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

Step 3

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- x$ .

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

Step 5

Apply the distributive property.

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

Step 6

Simplify.

Multiply $- 2$ by $3$ .

$f (x) = (3 x^{2} - 6 x + 3 \cdot 1) (x^{2} + 1)$

Multiply $3$ by $1$ .

$f (x) = (3 x^{2} - 6 x + 3) (x^{2} + 1)$

Step 7

Expand $(3 x^{2} - 6 x + 3) (x^{2} + 1)$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = 3 x^{2} x^{2} + 3 x^{2} \cdot 1 - 6 x \cdot x^{2} - 6 x \cdot 1 + 3 x^{2} + 3 \cdot 1$

Step 8

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$f (x) = 3 (x^{2} x^{2}) + 3 x^{2} \cdot 1 - 6 x \cdot x^{2} - 6 x \cdot 1 + 3 x^{2} + 3 \cdot 1$

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

$f (x) = 3 x^{2 + 2} + 3 x^{2} \cdot 1 - 6 x \cdot x^{2} - 6 x \cdot 1 + 3 x^{2} + 3 \cdot 1$

Add $2$ and $2$ .

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

Multiply $3$ by $1$ .

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 6$ by $1$ .

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

Multiply $3$ by $1$ .

$f (x) = 3 x^{4} + 3 x^{2} - 6 x^{3} - 6 x + 3 x^{2} + 3$

Simplify by adding terms.

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

$f (x) = 3 x^{4} + 6 x^{2} - 6 x^{3} - 6 x + 3$

Move $6 x^{2}$ .

$f (x) = 3 x^{4} - 6 x^{3} + 6 x^{2} - 6 x + 3$

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

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