Find the Roots/Zeros Using the Rational Roots Test f(x)=2x^3+7x^2+98x+343

$f (x) = 2 x^{3} + 7 x^{2} + 98 x + 343$

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.

$p = \pm 1, \pm 7, \pm 49, \pm 343$

$q = \pm 1, \pm 2$

Step 2

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

$\pm 1, \pm \frac{1}{2}, \pm 7, \pm \frac{7}{2}, \pm 49, \pm \frac{49}{2}, \pm 343, \pm \frac{343}{2}$

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.

$2 {(- \frac{7}{2})}^{3} + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Step 4

Simplify the expression. In this case, the expression is equal to $0$ so $x = - \frac{7}{2}$ is a root of the polynomial.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7}{2}$ .

$2 ({(- 1)}^{3} {(\frac{7}{2})}^{3}) + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Apply the product rule to $\frac{7}{2}$ .

$2 ({(- 1)}^{3} \frac{7^{3}}{2^{3}}) + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

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

$2 (- \frac{7^{3}}{2^{3}}) + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

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

$2 (- \frac{343}{2^{3}}) + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Raise $2$ to the power of $3$ .

$2 (- \frac{343}{8}) + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{343}{8}$ into the numerator.

$2 (\frac{- 343}{8}) + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Factor $2$ out of $8$ .

$2 \frac{- 343}{2 (4)} + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Cancel the common factor.

$2 \frac{- 343}{2 \cdot 4} + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Rewrite the expression.

$\frac{- 343}{4} + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Move the negative in front of the fraction.

$- \frac{343}{4} + 7 {(- \frac{7}{2})}^{2} + 98 (- \frac{7}{2}) + 343$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{7}{2}$ .

$- \frac{343}{4} + 7 ({(- 1)}^{2} {(\frac{7}{2})}^{2}) + 98 (- \frac{7}{2}) + 343$

Apply the product rule to $\frac{7}{2}$ .

$- \frac{343}{4} + 7 ({(- 1)}^{2} \frac{7^{2}}{2^{2}}) + 98 (- \frac{7}{2}) + 343$

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

$- \frac{343}{4} + 7 (1 \frac{7^{2}}{2^{2}}) + 98 (- \frac{7}{2}) + 343$

Multiply $\frac{7^{2}}{2^{2}}$ by $1$ .

$- \frac{343}{4} + 7 \frac{7^{2}}{2^{2}} + 98 (- \frac{7}{2}) + 343$

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

$- \frac{343}{4} + 7 \frac{49}{2^{2}} + 98 (- \frac{7}{2}) + 343$

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

$- \frac{343}{4} + 7 (\frac{49}{4}) + 98 (- \frac{7}{2}) + 343$

Multiply $7 (\frac{49}{4})$ .

Combine $7$ and $\frac{49}{4}$ .

$- \frac{343}{4} + \frac{7 \cdot 49}{4} + 98 (- \frac{7}{2}) + 343$

Multiply $7$ by $49$ .

$- \frac{343}{4} + \frac{343}{4} + 98 (- \frac{7}{2}) + 343$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$- \frac{343}{4} + \frac{343}{4} + 98 (\frac{- 7}{2}) + 343$

Factor $2$ out of $98$ .

$- \frac{343}{4} + \frac{343}{4} + 2 (49) \frac{- 7}{2} + 343$

Cancel the common factor.

$- \frac{343}{4} + \frac{343}{4} + 2 \cdot 49 \frac{- 7}{2} + 343$

Rewrite the expression.

$- \frac{343}{4} + \frac{343}{4} + 49 \cdot - 7 + 343$

Multiply $49$ by $- 7$ .

$- \frac{343}{4} + \frac{343}{4} - 343 + 343$

Combine fractions.

Combine the numerators over the common denominator.

$- 343 + 343 + \frac{- 343 + 343}{4}$

Add $- 343$ and $343$ .

$- 343 + 343 + \frac{0}{4}$

Find the common denominator.

Write $- 343$ as a fraction with denominator $1$ .

$\frac{- 343}{1} + 343 + \frac{0}{4}$

Multiply $\frac{- 343}{1}$ by $\frac{4}{4}$ .

$\frac{- 343}{1} \cdot \frac{4}{4} + 343 + \frac{0}{4}$

Multiply $\frac{- 343}{1}$ by $\frac{4}{4}$ .

$\frac{- 343 \cdot 4}{4} + 343 + \frac{0}{4}$

Write $343$ as a fraction with denominator $1$ .

$\frac{- 343 \cdot 4}{4} + \frac{343}{1} + \frac{0}{4}$

Multiply $\frac{343}{1}$ by $\frac{4}{4}$ .

$\frac{- 343 \cdot 4}{4} + \frac{343}{1} \cdot \frac{4}{4} + \frac{0}{4}$

Multiply $\frac{343}{1}$ by $\frac{4}{4}$ .

$\frac{- 343 \cdot 4}{4} + \frac{343 \cdot 4}{4} + \frac{0}{4}$

Combine the numerators over the common denominator.

$\frac{- 343 \cdot 4 + 343 \cdot 4}{4}$

Simplify each term.

Multiply $- 343$ by $4$ .

$\frac{- 1372 + 343 \cdot 4}{4}$

Multiply $343$ by $4$ .

$\frac{- 1372 + 1372}{4}$

Simplify the expression.

Add $- 1372$ and $1372$ .

$\frac{0}{4}$

Divide $0$ by $4$ .

$0$

Step 5

Since $- \frac{7}{2}$ is a known root, divide the polynomial by $x + \frac{7}{2}$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} + 7 x^{2} + 98 x + 343}{x + \frac{7}{2}}$

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.

$- \frac{7}{2}$	$2$	$7$	$98$	$343$
	\xa0			\xa0

The first number in the dividend $(2)$ is put into the first position of the result area (below the horizontal line).

$- \frac{7}{2}$	$2$	$7$	$98$	$343$
	\xa0			\xa0
	$2$

Multiply the newest entry in the result $(2)$ by the divisor $(- \frac{7}{2})$ and place the result of $(- 7)$ under the next term in the dividend $(7)$ .

$- \frac{7}{2}$	$2$	$7$	$98$	$343$
	\xa0	$- 7$		\xa0
	$2$

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

$- \frac{7}{2}$	$2$	$7$	$98$	$343$
	\xa0	$- 7$		\xa0
	$2$	$0$

Multiply the newest entry in the result $(0)$ by the divisor $(- \frac{7}{2})$ and place the result of $(0)$ under the next term in the dividend $(98)$ .

$- \frac{7}{2}$	$2$	$7$	$98$	$343$
	\xa0	$- 7$	$0$	\xa0
	$2$	$0$

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

$- \frac{7}{2}$	$2$	$7$	$98$	$343$
	\xa0	$- 7$	$0$	\xa0
	$2$	$0$	$98$

Multiply the newest entry in the result $(98)$ by the divisor $(- \frac{7}{2})$ and place the result of $(- 343)$ under the next term in the dividend $(343)$ .

$- \frac{7}{2}$	$2$	$7$	$98$	$343$
	\xa0	$- 7$	$0$	$- 343$
	$2$	$0$	$98$

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

$- \frac{7}{2}$	$2$	$7$	$98$	$343$
	\xa0	$- 7$	$0$	$- 343$
	$2$	$0$	$98$	$0$

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

$2 x^{2} + 0 x + 98$

Simplify the quotient polynomial.

$2 x^{2} + 98$

Step 7

Factor $2$ out of $2 x^{2} + 98$ .

Factor $2$ out of $2 x^{2}$ .

$(x + \frac{7}{2}) (2 (x^{2}) + 98)$

Factor $2$ out of $98$ .

$(x + \frac{7}{2}) (2 x^{2} + 2 \cdot 49)$

Factor $2$ out of $2 x^{2} + 2 \cdot 49$ .

$(x + \frac{7}{2}) (2 (x^{2} + 49))$

Step 8

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{3} + 7 x^{2}) + 98 x + 343 = 0$

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

$x^{2} (2 x + 7) + 49 (2 x + 7) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 7$ .

$(2 x + 7) (x^{2} + 49) = 0$

Step 9

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

$2 x + 7 = 0$

$x^{2} + 49 = 0$

Step 10

Set $2 x + 7$ equal to $0$ and solve for $x$ .

Set $2 x + 7$ equal to $0$ .

$2 x + 7 = 0$

Solve $2 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$2 x = - 7$

Divide each term in $2 x = - 7$ by $2$ and simplify.

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 11

Set $x^{2} + 49$ equal to $0$ and solve for $x$ .

Set $x^{2} + 49$ equal to $0$ .

$x^{2} + 49 = 0$

Solve $x^{2} + 49 = 0$ for $x$ .

Subtract $49$ from both sides of the equation.

$x^{2} = - 49$

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

$x = \pm \sqrt{- 49}$

Simplify $\pm \sqrt{- 49}$ .

Rewrite $- 49$ as $- 1 (49)$ .

$x = \pm \sqrt{- 1 (49)}$

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{49}$

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

$x = \pm i \cdot \sqrt{49}$

Rewrite $49$ as $7^{2}$ .

$x = \pm i \cdot \sqrt{7^{2}}$

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

$x = \pm i \cdot 7$

Move $7$ to the left of $i$ .

$x = \pm 7 i$

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.

$x = 7 i$

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

$x = - 7 i$

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

$x = 7 i, - 7 i$

Step 12

The final solution is all the values that make $(2 x + 7) (x^{2} + 49) = 0$ true.

$x = - \frac{7}{2}, 7 i, - 7 i$

Step 13

Find the Roots/Zeros Using the Rational Roots Test f(x)=2x^3+7x^2+98x+343

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