3x^3-9x^2-31x+5=0
How do I find the zeros, using synthetic division
let f(x) = 3x^3-9x^2-31x+5
f(1) = 3 - 9 - 31 + 5 ≠ 0
f(-1) ≠ 0
f(5) = 375 - 225 - 155 + 5 = 0
so x = 5 , and x-5 is a factor
I assume you know how to perform synthetic division.
You should get (x-5)(3x^2 + 6x - 1)
This quadratic produces two more zeros, use the quadratic formula to find them
There is a nice online calculator at
https://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php
It will show the steps involved.
To find the zeros of the given polynomial using synthetic division, you would follow these steps:
Step 1: Arrange the polynomial in descending order of powers.
Given polynomial: 3x^3 - 9x^2 - 31x + 5 = 0
Step 2: Identify the possible rational roots.
Since the coefficients are integers, the possible rational roots can be determined using the Rational Root Theorem. According to this theorem, the possible rational roots are the factors of the constant term (in this case, 5) divided by the factors of the leading coefficient (in this case, 3).
Possible rational roots (factors of 5): ±1, ±5
Possible rational roots (factors of 3): ±1, ±3
Potential rational roots: ±1, ±5, ±1/3, ±5/3
Step 3: Choose a potential rational root and perform synthetic division.
For synthetic division, you need to choose one potential rational root at a time and perform division to see if it gives a remainder of zero, indicating that it is a root of the polynomial.
Let's perform the synthetic division using one of the potential rational roots, say x = 1.
Coefficients: 3, -9, -31, 5
Divisor: x - 1
1 | 3 -9 -31 5
| 3 -6 -37
|__________________
3 -6 -37 -32
The result of synthetic division yields the coefficients of the quotient polynomial. In this case, the quotient polynomial is 3x^2 - 6x - 37, and the constant term is -32.
Step 4: Repeat Step 3 until a remainder of zero is obtained.
In this case, the remainder is not zero, indicating that x = 1 is not a root of the polynomial.
Repeat Step 3 with the remaining potential rational roots until a remainder of zero is obtained.
Step 5: Identify the roots of the polynomial.
If synthetic division results in a remainder of zero, it indicates that the chosen potential rational root is a zero of the polynomial.
In this case, none of the potential rational roots (1, 5, -1, -5, 1/3, -1/3, 5/3, -5/3) gives a remainder of zero. Therefore, the polynomial does not have any rational roots.
To find the roots of this polynomial, you can use numerical or graphical methods, such as factoring, using the quadratic formula, or using a calculator or computer software capable of finding the roots of higher-degree polynomials.