Write as a Set of Linear Factors x^3-11x^2+21x+49

$x^{3} - 11 x^{2} + 21 x + 49$

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.

$p = \pm 1, \pm 49, \pm 7 q = \pm 1$

Step 2

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

$\pm 1, \pm 49, \pm 7$

Step 3

Substitute $7$ and simplify the expression. In this case, the expression is equal to $0$ so $7$ is a root of the polynomial.

Substitute $7$ into the polynomial.

$7^{3} - 11 \cdot 7^{2} + 21 \cdot 7 + 49$

Raise $7$ to the power of $3$ .

$343 - 11 \cdot 7^{2} + 21 \cdot 7 + 49$

Raise $7$ to the power of $2$ .

$343 - 11 \cdot 49 + 21 \cdot 7 + 49$

Multiply $- 11$ by $49$ .

$343 - 539 + 21 \cdot 7 + 49$

Subtract $539$ from $343$ .

$- 196 + 21 \cdot 7 + 49$

Multiply $21$ by $7$ .

$- 196 + 147 + 49$

Add $- 196$ and $147$ .

$- 49 + 49$

Add $- 49$ and $49$ .

$0$

Step 4

Since $7$ is a known root, divide the polynomial by $x - 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 11 x^{2} + 21 x + 49}{x - 7}$

Step 5

Divide $x^{3} - 11 x^{2} + 21 x + 49$ by $x - 7$ .

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

$x$

$7$

$x^{3}$

$11 x^{2}$

$21 x$

$49$

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

					$x^{2}$
	$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			+	$x^{3}$	-	$7 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 7 x^{2}$

				$x^{2}$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$

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

				$x^{2}$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$

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

				$x^{2}$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$

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

				$x^{2}$	-	$4 x$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$
					-	$4 x^{2}$	+	$28 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 28 x$

				$x^{2}$	-	$4 x$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$
					+	$4 x^{2}$	-	$28 x$

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

				$x^{2}$	-	$4 x$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$
					+	$4 x^{2}$	-	$28 x$
							-	$7 x$

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

				$x^{2}$	-	$4 x$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$
					+	$4 x^{2}$	-	$28 x$
							-	$7 x$	+	$49$

Divide the highest order term in the dividend $- 7 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$4 x$	-	$7$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$
					+	$4 x^{2}$	-	$28 x$
							-	$7 x$	+	$49$

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$4 x$	-	$7$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$
					+	$4 x^{2}$	-	$28 x$
							-	$7 x$	+	$49$
							-	$7 x$	+	$49$

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x + 49$

				$x^{2}$	-	$4 x$	-	$7$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$
					+	$4 x^{2}$	-	$28 x$
							-	$7 x$	+	$49$
							+	$7 x$	-	$49$

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

				$x^{2}$	-	$4 x$	-	$7$
$x$	-	$7$		$x^{3}$	-	$11 x^{2}$	+	$21 x$	+	$49$
			-	$x^{3}$	+	$7 x^{2}$
					-	$4 x^{2}$	+	$21 x$
					+	$4 x^{2}$	-	$28 x$
							-	$7 x$	+	$49$
							+	$7 x$	-	$49$
										$0$

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

$x^{2} - 4 x - 7$

Step 6

Write $x^{3} - 11 x^{2} + 21 x + 49$ as a set of factors.

$(x - 7) (x^{2} - 4 x - 7)$

Write as a Set of Linear Factors x^3-11x^2+21x+49

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