Find the Roots/Zeros Using the Rational Roots Test a(x)=x^3-4x^2-7x+28

$a (x) = x^{3} - 4 x^{2} - 7 x + 28$

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.

$p = \pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28$

$q = \pm 1$

Step 2

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

$\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28$

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.

${(4)}^{3} - 4 {(4)}^{2} - 7 \cdot 4 + 28$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = 4$ is a root of the polynomial.

Simplify each term.

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

$64 - 4 {(4)}^{2} - 7 \cdot 4 + 28$

Raise $4$ to the power of $2$ .

$64 - 4 \cdot 16 - 7 \cdot 4 + 28$

Multiply $- 4$ by $16$ .

$64 - 64 - 7 \cdot 4 + 28$

Multiply $- 7$ by $4$ .

$64 - 64 - 28 + 28$

Simplify by adding and subtracting.

Subtract $64$ from $64$ .

$0 - 28 + 28$

Subtract $28$ from $0$ .

$- 28 + 28$

Add $- 28$ and $28$ .

$0$

Step 5

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

$\frac{x^{3} - 4 x^{2} - 7 x + 28}{x - 4}$

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.

$4$	$1$	$- 4$	$- 7$	$28$
	\xa0			\xa0

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

$4$	$1$	$- 4$	$- 7$	$28$
	\xa0			\xa0
	$1$

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

$4$	$1$	$- 4$	$- 7$	$28$
	\xa0	$4$		\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.

$4$	$1$	$- 4$	$- 7$	$28$
	\xa0	$4$		\xa0
	$1$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(4)$ and place the result of $(0)$ under the next term in the dividend $(- 7)$ .

$4$	$1$	$- 4$	$- 7$	$28$
	\xa0	$4$	$0$	\xa0
	$1$	$0$

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

$4$	$1$	$- 4$	$- 7$	$28$
	\xa0	$4$	$0$	\xa0
	$1$	$0$	$- 7$

Multiply the newest entry in the result $(- 7)$ by the divisor $(4)$ and place the result of $(- 28)$ under the next term in the dividend $(28)$ .

$4$	$1$	$- 4$	$- 7$	$28$
	\xa0	$4$	$0$	$- 28$
	$1$	$0$	$- 7$

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

$4$	$1$	$- 4$	$- 7$	$28$
	\xa0	$4$	$0$	$- 28$
	$1$	$0$	$- 7$	$0$

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

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

Simplify the quotient polynomial.

$x^{2} - 7$

Step 7

Solve the equation to find any remaining roots.

Add $7$ to both sides of the equation.

$x^{2} = 7$

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

$x = \pm \sqrt{7}$

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.

$x = \sqrt{7}$

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

$x = - \sqrt{7}$

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

$x = \sqrt{7}, - \sqrt{7}$

Step 8

The polynomial can be written as a set of linear factors.

$(x - 4) (x - \sqrt{7}) (x + \sqrt{7})$

Step 9

These are the roots (zeros) of the polynomial $x^{3} - 4 x^{2} - 7 x + 28$ .

$x = 4, \sqrt{7}, - \sqrt{7}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 4, \sqrt{7}, - \sqrt{7}$

Decimal Form:

$x = 4, 2.64575131 \dots, - 2.64575131 \dots$

Step 11

Find the Roots/Zeros Using the Rational Roots Test a(x)=x^3-4x^2-7x+28

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