Write as a Set of Linear Factors 4x^3-27x^2-11x-1=0

$4 x^{3} - 27 x^{2} - 11 x - 1 = 0$

Step 1

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.

$p = \pm 1$

$q = \pm 1, \pm 4, \pm 2 = 0$

Step 2

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

$\pm 1, \pm 0.25, \pm 0.5 = 0$

Step 3

Substitute $- 0.25$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.25$ is a root of the polynomial.

Substitute $- 0.25$ into the polynomial.

$4 {(- 0.25)}^{3} - 27 {(- 0.25)}^{2} - 11 \cdot - 0.25 - 1 = 0$

Raise $- 0.25$ to the power of $3$ .

$4 \cdot - 0.015625 - 27 {(- 0.25)}^{2} - 11 \cdot - 0.25 - 1 = 0$

Multiply $4$ by $- 0.015625$ .

$- 0.0625 - 27 {(- 0.25)}^{2} - 11 \cdot - 0.25 - 1 = 0$

Raise $- 0.25$ to the power of $2$ .

$- 0.0625 - 27 \cdot 0.0625 - 11 \cdot - 0.25 - 1 = 0$

Multiply $- 27$ by $0.0625$ .

$- 0.0625 - 1.6875 - 11 \cdot - 0.25 - 1 = 0$

Subtract $1.6875$ from $- 0.0625$ .

$- 1.75 - 11 \cdot - 0.25 - 1 = 0$

Multiply $- 11$ by $- 0.25$ .

$- 1.75 + 2.75 - 1 = 0$

Add $- 1.75$ and $2.75$ .

$1 - 1 = 0$

Subtract $1$ from $1$ .

$0 = 0$

Step 4

Since $- 0.25$ is a known root, divide the polynomial by $4 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 27 x^{2} - 11 x - 1}{4 x + 1} = 0$

Step 5

Divide $4 x^{3} - 27 x^{2} - 11 x - 1$ by $4 x + 1$ .

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

$= 0$

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $4 x$ .

$= 0$

Multiply the new quotient term by the divisor.

$= 0$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} + x^{2}$

$= 0$

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

$= 0$

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

$= 0$

Divide the highest order term in the dividend $- 28 x^{2}$ by the highest order term in divisor $4 x$ .

$= 0$

Multiply the new quotient term by the divisor.

$= 0$

The expression needs to be subtracted from the dividend, so change all the signs in $- 28 x^{2} - 7 x$

$= 0$

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

$= 0$

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

$= 0$

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $4 x$ .

$= 0$

Multiply the new quotient term by the divisor.

$= 0$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 1$

$= 0$

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

$= 0$

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

$x^{2} - 7 x - 1 = 0$

Step 6

Write $4 x^{3} - 27 x^{2} - 11 x - 1$ as a set of factors.

$(4 x + 1) (x^{2} - 7 x - 1) = 0$

Write as a Set of Linear Factors 4x^3-27x^2-11x-1=0

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