Find the Roots/Zeros Using the Rational Roots Test x^4-12x^3+15x^2-48x+44

$x^{4} - 12 x^{3} + 15 x^{2} - 48 x + 44$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 11, \pm 22, \pm 44$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 4, \pm 11, \pm 22, \pm 44$

Step 3

Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is $0$ , which means it is a root.

${(1)}^{4} - 12 {(1)}^{3} + 15 {(1)}^{2} - 48 \cdot 1 + 44$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 1$ is a root of the polynomial.

Simplify each term.

One to any power is one.

$1 - 12 {(1)}^{3} + 15 {(1)}^{2} - 48 \cdot 1 + 44$

One to any power is one.

$1 - 12 \cdot 1 + 15 {(1)}^{2} - 48 \cdot 1 + 44$

Multiply $- 12$ by $1$ .

$1 - 12 + 15 {(1)}^{2} - 48 \cdot 1 + 44$

One to any power is one.

$1 - 12 + 15 \cdot 1 - 48 \cdot 1 + 44$

Multiply $15$ by $1$ .

$1 - 12 + 15 - 48 \cdot 1 + 44$

Multiply $- 48$ by $1$ .

$1 - 12 + 15 - 48 + 44$

Simplify by adding and subtracting.

Subtract $12$ from $1$ .

$- 11 + 15 - 48 + 44$

Add $- 11$ and $15$ .

$4 - 48 + 44$

Subtract $48$ from $4$ .

$- 44 + 44$

Add $- 44$ and $44$ .

$0$

Step 5

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

$\frac{x^{4} - 12 x^{3} + 15 x^{2} - 48 x + 44}{x - 1}$

Step 6

Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by $1$ .

Place the numbers representing the divisor and the dividend into a division-like configuration.

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0				\xa0

The first number in the dividend $(1)$ is put into the first position of the result area (below the horizontal line).

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0				\xa0
	$1$

Multiply the newest entry in the result $(1)$ by the divisor $(1)$ and place the result of $(1)$ under the next term in the dividend $(- 12)$ .

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0	$1$			\xa0
	$1$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0	$1$			\xa0
	$1$	$- 11$

Multiply the newest entry in the result $(- 11)$ by the divisor $(1)$ and place the result of $(- 11)$ under the next term in the dividend $(15)$ .

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0	$1$	$- 11$		\xa0
	$1$	$- 11$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0	$1$	$- 11$		\xa0
	$1$	$- 11$	$4$

Multiply the newest entry in the result $(4)$ by the divisor $(1)$ and place the result of $(4)$ under the next term in the dividend $(- 48)$ .

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0	$1$	$- 11$	$4$	\xa0
	$1$	$- 11$	$4$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0	$1$	$- 11$	$4$	\xa0
	$1$	$- 11$	$4$	$- 44$

Multiply the newest entry in the result $(- 44)$ by the divisor $(1)$ and place the result of $(- 44)$ under the next term in the dividend $(44)$ .

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0	$1$	$- 11$	$4$	$- 44$
	$1$	$- 11$	$4$	$- 44$

Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.

$1$	$1$	$- 12$	$15$	$- 48$	$44$
	\xa0	$1$	$- 11$	$4$	$- 44$
	$1$	$- 11$	$4$	$- 44$	$0$

All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.

$1 x^{3} + - 11 x^{2} + (4) x - 44$

Simplify the quotient polynomial.

$x^{3} - 11 x^{2} + 4 x - 44$

Step 7

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(x - 1) (x^{3} - 11 x^{2}) + 4 x - 44$

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

$(x - 1) + x^{2} (x - 11) + 4 (x - 11)$

$(x - 1) x^{2} (x - 11) + 4 (x - 11)$

Step 8

Factor the polynomial by factoring out the greatest common factor, $x - 11$ .

$(x - 1) (x - 11) (x^{2} + 4)$

Step 9

Factor the left side of the equation.

Regroup terms.

$- 12 x^{3} - 48 x + x^{4} + 15 x^{2} + 44 = 0$

Factor $- 12 x$ out of $- 12 x^{3} - 48 x$ .

Factor $- 12 x$ out of $- 12 x^{3}$ .

$- 12 x \cdot x^{2} - 48 x + x^{4} + 15 x^{2} + 44 = 0$

Factor $- 12 x$ out of $- 48 x$ .

$- 12 x \cdot x^{2} - 12 x \cdot 4 + x^{4} + 15 x^{2} + 44 = 0$

Factor $- 12 x$ out of $- 12 x (x^{2}) - 12 x (4)$ .

$- 12 x (x^{2} + 4) + x^{4} + 15 x^{2} + 44 = 0$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 12 x (x^{2} + 4) + {(x^{2})}^{2} + 15 x^{2} + 44 = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 12 x (x^{2} + 4) + u^{2} + 15 u + 44 = 0$

Factor $u^{2} + 15 u + 44$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $44$ and whose sum is $15$ .

$4, 11$

Write the factored form using these integers.

$- 12 x (x^{2} + 4) + (u + 4) (u + 11) = 0$

Replace all occurrences of $u$ with $x^{2}$ .

$- 12 x (x^{2} + 4) + (x^{2} + 4) (x^{2} + 11) = 0$

Factor $x^{2} + 4$ out of $- 12 x (x^{2} + 4) + (x^{2} + 4) (x^{2} + 11)$ .

Factor $x^{2} + 4$ out of $- 12 x (x^{2} + 4)$ .

$(x^{2} + 4) (- 12 x) + (x^{2} + 4) (x^{2} + 11) = 0$

Factor $x^{2} + 4$ out of $(x^{2} + 4) (- 12 x) + (x^{2} + 4) (x^{2} + 11)$ .

$(x^{2} + 4) (- 12 x + x^{2} + 11) = 0$

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x^{2} + 4) (- 12 u + u^{2} + 11) = 0$

Factor $- 12 u + u^{2} + 11$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $11$ and whose sum is $- 12$ .

$- 11, - 1$

Write the factored form using these integers.

$(x^{2} + 4) ((u - 11) (u - 1)) = 0$

Factor.

Replace all occurrences of $u$ with $x$ .

$(x^{2} + 4) ((x - 11) (x - 1)) = 0$

Remove unnecessary parentheses.

$(x^{2} + 4) (x - 11) (x - 1) = 0$

Step 10

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} + 4 = 0$

$x - 11 = 0$

$x - 1 = 0$

Step 11

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

Set $x^{2} + 4$ equal to $0$ .

$x^{2} + 4 = 0$

Solve $x^{2} + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x = \pm 2 i$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i, - 2 i$

Step 12

Set $x - 11$ equal to $0$ and solve for $x$ .

Set $x - 11$ equal to $0$ .

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

Step 13

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 14

The final solution is all the values that make $(x^{2} + 4) (x - 11) (x - 1) = 0$ true.

$x = 2 i, - 2 i, 11, 1$

Step 15

Find the Roots/Zeros Using the Rational Roots Test x^4-12x^3+15x^2-48x+44

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