Find the Roots (Zeros) f(x)=x^3+3x^2-9x+5

$f (x) = x^{3} + 3 x^{2} - 9 x + 5$

Step 1

Set $x^{3} + 3 x^{2} - 9 x + 5$ equal to $0$ .

$x^{3} + 3 x^{2} - 9 x + 5 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{3} + 3 x^{2} - 9 x + 5$ 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.

$p = \pm 1, \pm 5 q = \pm 1$

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

$\pm 1, \pm 5$

Substitute $1$ and simplify the expression. In this case, the expression is equal to $0$ so $1$ is a root of the polynomial.

Substitute $1$ into the polynomial.

$1^{3} + 3 \cdot 1^{2} - 9 \cdot 1 + 5$

Raise $1$ to the power of $3$ .

$1 + 3 \cdot 1^{2} - 9 \cdot 1 + 5$

Raise $1$ to the power of $2$ .

$1 + 3 \cdot 1 - 9 \cdot 1 + 5$

Multiply $3$ by $1$ .

$1 + 3 - 9 \cdot 1 + 5$

Add $1$ and $3$ .

$4 - 9 \cdot 1 + 5$

Multiply $- 9$ by $1$ .

$4 - 9 + 5$

Subtract $9$ from $4$ .

$- 5 + 5$

Add $- 5$ and $5$ .

$0$

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.

$\frac{x^{3} + 3 x^{2} - 9 x + 5}{x - 1}$

Divide $x^{3} + 3 x^{2} - 9 x + 5$ by $x - 1$ .

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

$x$

$1$

$x^{3}$

$3 x^{2}$

$9 x$

$5$

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

					$x^{2}$
	$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			+	$x^{3}$	-	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$

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

				$x^{2}$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$

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

				$x^{2}$	+	$4 x$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$
					+	$4 x^{2}$	-	$4 x$

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

				$x^{2}$	+	$4 x$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$
					-	$4 x^{2}$	+	$4 x$

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

				$x^{2}$	+	$4 x$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$
					-	$4 x^{2}$	+	$4 x$
							-	$5 x$

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

				$x^{2}$	+	$4 x$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$
					-	$4 x^{2}$	+	$4 x$
							-	$5 x$	+	$5$

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$4 x$	-	$5$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$
					-	$4 x^{2}$	+	$4 x$
							-	$5 x$	+	$5$

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$4 x$	-	$5$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$
					-	$4 x^{2}$	+	$4 x$
							-	$5 x$	+	$5$
							-	$5 x$	+	$5$

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x + 5$

				$x^{2}$	+	$4 x$	-	$5$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$
					-	$4 x^{2}$	+	$4 x$
							-	$5 x$	+	$5$
							+	$5 x$	-	$5$

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

				$x^{2}$	+	$4 x$	-	$5$
$x$	-	$1$		$x^{3}$	+	$3 x^{2}$	-	$9 x$	+	$5$
			-	$x^{3}$	+	$x^{2}$
					+	$4 x^{2}$	-	$9 x$
					-	$4 x^{2}$	+	$4 x$
							-	$5 x$	+	$5$
							+	$5 x$	-	$5$
										$0$

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

$x^{2} + 4 x - 5$

Write $x^{3} + 3 x^{2} - 9 x + 5$ as a set of factors.

$(x - 1) (x^{2} + 4 x - 5) = 0$

Factor $x^{2} + 4 x - 5$ 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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Write the factored form using these integers.

$(x - 1) ((x - 1) (x + 5)) = 0$

Remove unnecessary parentheses.

$(x - 1) (x - 1) (x + 5) = 0$

Combine like factors.

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

$(x - 1) (x - 1) (x + 5) = 0$

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

$(x - 1) (x - 1) (x + 5) = 0$

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

${(x - 1)}^{1 + 1} (x + 5) = 0$

Add $1$ and $1$ .

${(x - 1)}^{2} (x + 5) = 0$

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

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

$x + 5 = 0$

Set ${(x - 1)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{2}$ equal to $0$ .

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

Solve ${(x - 1)}^{2} = 0$ for $x$ .

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make ${(x - 1)}^{2} (x + 5) = 0$ true.

$x = 1, - 5$

Step 3

Find the Roots (Zeros) f(x)=x^3+3x^2-9x+5

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