Write in Standard Form 3x(x-7)(x-1)^2

$3 x (x - 7) {(x - 1)}^{2}$

Step 1

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

$a x^{2} + b x + c$

Step 2

Apply the distributive property.

$(3 x \cdot x + 3 x \cdot - 7) {(x - 1)}^{2}$

Step 3

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

Move $x$ .

$(3 (x \cdot x) + 3 x \cdot - 7) {(x - 1)}^{2}$

Multiply $x$ by $x$ .

$(3 x^{2} + 3 x \cdot - 7) {(x - 1)}^{2}$

Step 4

Simplify the expression.

Multiply $- 7$ by $3$ .

$(3 x^{2} - 21 x) {(x - 1)}^{2}$

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

$(3 x^{2} - 21 x) ((x - 1) (x - 1))$

Step 5

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

Apply the distributive property.

$(3 x^{2} - 21 x) (x (x - 1) - 1 (x - 1))$

Apply the distributive property.

$(3 x^{2} - 21 x) (x \cdot x + x \cdot - 1 - 1 (x - 1))$

Apply the distributive property.

$(3 x^{2} - 21 x) (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$(3 x^{2} - 21 x) (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)$

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

$(3 x^{2} - 21 x) (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)$

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

$(3 x^{2} - 21 x) (x^{2} - x - 1 x - 1 \cdot - 1)$

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

$(3 x^{2} - 21 x) (x^{2} - x - x - 1 \cdot - 1)$

Multiply $- 1$ by $- 1$ .

$(3 x^{2} - 21 x) (x^{2} - x - x + 1)$

Subtract $x$ from $- x$ .

$(3 x^{2} - 21 x) (x^{2} - 2 x + 1)$

Step 7

Expand $(3 x^{2} - 21 x) (x^{2} - 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{2} x^{2} + 3 x^{2} (- 2 x) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Step 8

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$3 (x^{2} x^{2}) + 3 x^{2} (- 2 x) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

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

$3 x^{2 + 2} + 3 x^{2} (- 2 x) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Add $2$ and $2$ .

$3 x^{4} + 3 x^{2} (- 2 x) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Rewrite using the commutative property of multiplication.

$3 x^{4} + 3 \cdot - 2 x^{2} x + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

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

Move $x$ .

$3 x^{4} + 3 \cdot - 2 (x \cdot x^{2}) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

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

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

$3 x^{4} + 3 \cdot - 2 (x^{1} x^{2}) + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

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

$3 x^{4} + 3 \cdot - 2 x^{1 + 2} + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Add $1$ and $2$ .

$3 x^{4} + 3 \cdot - 2 x^{3} + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Multiply $3$ by $- 2$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} \cdot 1 - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

Multiply $3$ by $1$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x \cdot x^{2} - 21 x (- 2 x) - 21 x \cdot 1$

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

Move $x^{2}$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 (x^{2} x) - 21 x (- 2 x) - 21 x \cdot 1$

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

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

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 (x^{2} x^{1}) - 21 x (- 2 x) - 21 x \cdot 1$

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

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{2 + 1} - 21 x (- 2 x) - 21 x \cdot 1$

Add $2$ and $1$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} - 21 x (- 2 x) - 21 x \cdot 1$

Rewrite using the commutative property of multiplication.

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} - 21 \cdot - 2 x \cdot x - 21 x \cdot 1$

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

Move $x$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} - 21 \cdot - 2 (x \cdot x) - 21 x \cdot 1$

Multiply $x$ by $x$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} - 21 \cdot - 2 x^{2} - 21 x \cdot 1$

Multiply $- 21$ by $- 2$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} + 42 x^{2} - 21 x \cdot 1$

Multiply $- 21$ by $1$ .

$3 x^{4} - 6 x^{3} + 3 x^{2} - 21 x^{3} + 42 x^{2} - 21 x$

Simplify by adding terms.

Subtract $21 x^{3}$ from $- 6 x^{3}$ .

$3 x^{4} - 27 x^{3} + 3 x^{2} + 42 x^{2} - 21 x$

Add $3 x^{2}$ and $42 x^{2}$ .

$3 x^{4} - 27 x^{3} + 45 x^{2} - 21 x$

Write in Standard Form 3x(x-7)(x-1)^2

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