# Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+5x^2-139x-143

Find the Roots/Zeros Using the Rational Roots Test f(x)=x^3+5x^2-139x-143
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.
Find every combination of . These are the possible roots of the polynomial function.
Substitute the possible roots one by one into the polynomial to find the actual roots. Simplify to check if the value is , which means it is a root.
Simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
Simplify each term.
Raise to the power of .
Raise to the power of .
Multiply by .
Multiply by .
Subtract from .
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.
Next, find the roots of the remaining polynomial. The order of the polynomial has been reduced by .
Place the numbers representing the divisor and the dividend into a division-like configuration.
The first number in the dividend is put into the first position of the result area (below the horizontal line).
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
Multiply the newest entry in the result by the divisor and place the result of under the next term in the dividend .
Add the product of the multiplication and the number from the dividend and put the result in the next position on the result line.
All numbers except the last become the coefficients of the quotient polynomial. The last value in the result line is the remainder.
Simplify the quotient polynomial.
Solve the equation to find any remaining roots.
Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
Simplify the expression to solve for the portion of the .
Simplify the numerator.
Raise to the power of .
Multiply by .
Multiply by .
Rewrite as .
Factor out of .
Rewrite as .
Pull terms out from under the radical.
Multiply by .
Simplify .
Change the to .
The final answer is the combination of both solutions.
The polynomial can be written as a set of linear factors.
These are the roots (zeros) of the polynomial .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
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### Name

Name one billion two hundred sixty-two million nine hundred eighty-five thousand twenty

### Interesting facts

• 1262985020 has 32 divisors, whose sum is 2896429536
• The reverse of 1262985020 is 0205892621
• Previous prime number is 1181

### Basic properties

• Is Prime? no
• Number parity even
• Number length 10
• Sum of Digits 35
• Digital Root 8

### Name

Name five hundred ninety-nine million three hundred seventy-seven thousand eight hundred seventy-two

### Interesting facts

• 599377872 has 64 divisors, whose sum is 4045800960
• The reverse of 599377872 is 278773995
• Previous prime number is 3

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 57
• Digital Root 3

### Name

Name three hundred seventy-seven million nine hundred eighty-seven thousand four hundred forty

### Interesting facts

• 377987440 has 128 divisors, whose sum is 2299464288
• The reverse of 377987440 is 044789773
• Previous prime number is 823

### Basic properties

• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 49
• Digital Root 4