Get zeroes of a polynomial when some are known

How to Find All Zeroes of a Polynomial When Some Are Known

If you are given a polynomial and some of its zeroes, you can use that information to find the remaining zeroes. Here’s how the method works using this polynomial:

Given Polynomial

P(x) = 2x⁴ - 3x³ - 3x² + 6x - 2

It is known that √2 and −√2 are two of the polynomial’s zeroes.

Step 1: Form a Quadratic Factor from Known Zeroes

Since √2 and −√2 are zeroes, their corresponding factor is:

(x - √2)(x + √2) = x² - 2

Step 2: Divide the Polynomial by x² - 2

We now divide P(x) by x² - 2 using polynomial division (or synthetic division if applicable):

Divide:
P(x) = 2x⁴ - 3x³ - 3x² + 6x - 2
By: x² - 2

Using polynomial division (or long division), we get:

Quotient: 2x² - 3x - 1

Step 3: Factor the Quotient Further

Next, we factor the obtained quadratic expression:
2x² - 3x - 1

To factor this expression, we can apply either the method of splitting the middle term or use the quadratic formula.

2x² - 3x - 1 = (2x + 1)(x - 1)

Step 4: Write All Zeroes

The zeroes of the polynomial are the values of x that make each factor zero:

• x = √2
• x = −√2
• x = −1/2
• x = 1

Given two zeroes of a polynomial, we factored the corresponding quadratic and divided the original polynomial by it. We then factored the resulting quotient to find all zeroes. Hence, the polynomial P(x) = 2x⁴ - 3x³ - 3x² + 6x - 2 has the following zeroes:

{√2, −√2, −1/2, 1}