Trigonometry Examples

Find the Roots (Zeros) f(x)=3x^3-16x^2-30x-12
Step 1
Set equal to .
Step 2
Solve for .
Tap for more steps...
Step 2.1
Factor using the rational roots test.
Tap for more steps...
Step 2.1.1
If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Step 2.1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 2.1.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
Tap for more steps...
Step 2.1.3.1
Substitute into the polynomial.
Step 2.1.3.2
Raise to the power of .
Step 2.1.3.3
Multiply by .
Step 2.1.3.4
Raise to the power of .
Step 2.1.3.5
Multiply by .
Step 2.1.3.6
Subtract from .
Step 2.1.3.7
Multiply by .
Step 2.1.3.8
Add and .
Step 2.1.3.9
Subtract from .
Step 2.1.4
Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
Step 2.1.5
Divide by .
Tap for more steps...
Step 2.1.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
+---
Step 2.1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
+---
Step 2.1.5.3
Multiply the new quotient term by the divisor.
+---
++
Step 2.1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
+---
--
Step 2.1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
+---
--
-
Step 2.1.5.6
Pull the next terms from the original dividend down into the current dividend.
+---
--
--
Step 2.1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
-
+---
--
--
Step 2.1.5.8
Multiply the new quotient term by the divisor.
-
+---
--
--
--
Step 2.1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
-
+---
--
--
++
Step 2.1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-
+---
--
--
++
-
Step 2.1.5.11
Pull the next terms from the original dividend down into the current dividend.
-
+---
--
--
++
--
Step 2.1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
--
+---
--
--
++
--
Step 2.1.5.13
Multiply the new quotient term by the divisor.
--
+---
--
--
++
--
--
Step 2.1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
--
+---
--
--
++
--
++
Step 2.1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
--
+---
--
--
++
--
++
Step 2.1.5.16
Since the remander is , the final answer is the quotient.
Step 2.1.6
Write as a set of factors.
Step 2.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.3
Set equal to and solve for .
Tap for more steps...
Step 2.3.1
Set equal to .
Step 2.3.2
Solve for .
Tap for more steps...
Step 2.3.2.1
Subtract from both sides of the equation.
Step 2.3.2.2
Divide each term in by and simplify.
Tap for more steps...
Step 2.3.2.2.1
Divide each term in by .
Step 2.3.2.2.2
Simplify the left side.
Tap for more steps...
Step 2.3.2.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.3.2.2.2.1.1
Cancel the common factor.
Step 2.3.2.2.2.1.2
Divide by .
Step 2.3.2.2.3
Simplify the right side.
Tap for more steps...
Step 2.3.2.2.3.1
Move the negative in front of the fraction.
Step 2.4
Set equal to and solve for .
Tap for more steps...
Step 2.4.1
Set equal to .
Step 2.4.2
Solve for .
Tap for more steps...
Step 2.4.2.1
Use the quadratic formula to find the solutions.
Step 2.4.2.2
Substitute the values , , and into the quadratic formula and solve for .
Step 2.4.2.3
Simplify.
Tap for more steps...
Step 2.4.2.3.1
Simplify the numerator.
Tap for more steps...
Step 2.4.2.3.1.1
Raise to the power of .
Step 2.4.2.3.1.2
Multiply .
Tap for more steps...
Step 2.4.2.3.1.2.1
Multiply by .
Step 2.4.2.3.1.2.2
Multiply by .
Step 2.4.2.3.1.3
Add and .
Step 2.4.2.3.1.4
Rewrite as .
Tap for more steps...
Step 2.4.2.3.1.4.1
Factor out of .
Step 2.4.2.3.1.4.2
Rewrite as .
Step 2.4.2.3.1.5
Pull terms out from under the radical.
Step 2.4.2.3.2
Multiply by .
Step 2.4.2.3.3
Simplify .
Step 2.4.2.4
The final answer is the combination of both solutions.
Step 2.5
The final solution is all the values that make true.
Step 3
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 4