Write as a Set of Linear Factors g(x)=x^4-6x^3+18x^2-54x+81

$g (x) = x^{4} - 6 x^{3} + 18 x^{2} - 54 x + 81$

Step 1

Regroup terms.

$g (x) = - 6 x^{3} + 18 x^{2} + x^{4} - 54 x + 81$

Step 2

Factor $- 6 x^{2}$ out of $- 6 x^{3} + 18 x^{2}$ .

Factor $- 6 x^{2}$ out of $- 6 x^{3}$ .

$g (x) = - 6 x^{2} x + 18 x^{2} + x^{4} - 54 x + 81$

Factor $- 6 x^{2}$ out of $18 x^{2}$ .

$g (x) = - 6 x^{2} x - 6 x^{2} \cdot - 3 + x^{4} - 54 x + 81$

Factor $- 6 x^{2}$ out of $- 6 x^{2} (x) - 6 x^{2} (- 3)$ .

$g (x) = - 6 x^{2} (x - 3) + x^{4} - 54 x + 81$

Step 3

Factor $x^{4} - 54 x + 81$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$g (x) = p = \pm 1, \pm 81, \pm 3, \pm 27, \pm 9$

$q = \pm 1$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$g (x) = \pm 1, \pm 81, \pm 3, \pm 27, \pm 9$

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

Substitute $3$ into the polynomial.

$g (x) = 3^{4} - 54 \cdot 3 + 81$

Raise $3$ to the power of $4$ .

$g (x) = 81 - 54 \cdot 3 + 81$

Multiply $- 54$ by $3$ .

$g (x) = 81 - 162 + 81$

Subtract $162$ from $81$ .

$g (x) = - 81 + 81$

Add $- 81$ and $81$ .

$g (x) = 0$

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$g (x) = \frac{x^{4} - 54 x + 81}{x - 3}$

Divide $x^{4} - 54 x + 81$ by $x - 3$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$g (x) =$

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$g (x) =$

Multiply the new quotient term by the divisor.

$g (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} - 3 x^{3}$

$g (x) =$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$g (x) =$

Pull the next terms from the original dividend down into the current dividend.

$g (x) =$

Divide the highest order term in the dividend $3 x^{3}$ by the highest order term in divisor $x$ .

$g (x) =$

Multiply the new quotient term by the divisor.

$g (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 9 x^{2}$

$g (x) =$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$g (x) =$

Pull the next terms from the original dividend down into the current dividend.

$g (x) =$

Divide the highest order term in the dividend $9 x^{2}$ by the highest order term in divisor $x$ .

$g (x) =$

Multiply the new quotient term by the divisor.

$g (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{2} - 27 x$

$g (x) =$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$g (x) =$

Pull the next terms from the original dividend down into the current dividend.

$g (x) =$

Divide the highest order term in the dividend $- 27 x$ by the highest order term in divisor $x$ .

$g (x) =$

Multiply the new quotient term by the divisor.

$g (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 27 x + 81$

$g (x) =$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$g (x) =$

Since the remander is $0$ , the final answer is the quotient.

$g (x) = x^{3} + 3 x^{2} + 9 x - 27$

Write $x^{4} - 54 x + 81$ as a set of factors.

$g (x) = - 6 x^{2} (x - 3) + (x - 3) (x^{3} + 3 x^{2} + 9 x - 27)$

Step 4

Factor $x - 3$ out of $- 6 x^{2} (x - 3) + (x - 3) (x^{3} + 3 x^{2} + 9 x - 27)$ .

Factor $x - 3$ out of $- 6 x^{2} (x - 3)$ .

$g (x) = (x - 3) (- 6 x^{2}) + (x - 3) (x^{3} + 3 x^{2} + 9 x - 27)$

Factor $x - 3$ out of $(x - 3) (- 6 x^{2}) + (x - 3) (x^{3} + 3 x^{2} + 9 x - 27)$ .

$g (x) = (x - 3) (- 6 x^{2} + x^{3} + 3 x^{2} + 9 x - 27)$

Step 5

Add $- 6 x^{2}$ and $3 x^{2}$ .

$g (x) = (x - 3) (x^{3} - 3 x^{2} + 9 x - 27)$

Step 6

Factor.

Rewrite $x^{3} - 3 x^{2} + 9 x - 27$ in a factored form.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$g (x) = (x - 3) ((x^{3} - 3 x^{2}) + 9 x - 27)$

Factor out the greatest common factor (GCF) from each group.

$g (x) = (x - 3) (x^{2} (x - 3) + 9 (x - 3))$

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$g (x) = (x - 3) ((x - 3) (x^{2} + 9))$

Remove unnecessary parentheses.

$g (x) = (x - 3) (x - 3) (x^{2} + 9)$

Step 7

Combine exponents.

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

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

Add $1$ and $1$ .

$g (x) = {(x - 3)}^{2} (x^{2} + 9)$

Write as a Set of Linear Factors g(x)=x^4-6x^3+18x^2-54x+81

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