You use synthetic division and the Remainder Theorem: if, after performing synthetic division with K, the remainder is 0, then K is a zero of the polynomial.
Here's a breakdown of the process:
Understanding the Remainder Theorem
The Remainder Theorem states that if you divide a polynomial f(x) by x - k, the remainder is equal to f(k). In simpler terms, if you plug k into the polynomial, the result is the same as the remainder you get from synthetic division.
Steps to Determine if K is a Zero Using Synthetic Division:
- Set up synthetic division: Write k to the left and the coefficients of the polynomial to the right, ensuring the polynomial is written in descending order of powers of x, and including 0 as a coefficient for any missing terms.
- Perform synthetic division:
- Bring down the first coefficient.
- Multiply the first coefficient by k and write the result below the second coefficient.
- Add the second coefficient and the result from the previous step.
- Repeat the multiplication and addition process for all remaining coefficients.
- Check the remainder: The last number in the bottom row is the remainder.
- Decision:
- If the remainder is 0: Then f(k) = 0, meaning k is a zero (or root) of the polynomial. x - k is also a factor of the polynomial.
- If the remainder is not 0: Then f(k) ≠ 0, meaning k is not a zero of the polynomial.
Example
Let's say we want to determine if k = 2 is a zero of the polynomial f(x) = x³ - 4x² + 5x - 2.
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Set up:
2 | 1 -4 5 -2 -
Perform synthetic division:
2 | 1 -4 5 -2 | 2 -4 2 ------------------ 1 -2 1 0 -
Check the remainder: The remainder is 0.
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Decision: Since the remainder is 0, k = 2 is a zero of the polynomial f(x) = x³ - 4x² + 5x - 2. Furthermore, x - 2 is a factor of the polynomial.
In Summary
Synthetic division offers a quick and efficient way to evaluate a polynomial at a specific value (k) and determine if that value is a zero of the polynomial. If the remainder after synthetic division is 0, then the given number k is indeed a zero of the polynomial.
