Synthetic division is a streamlined way to divide a polynomial by a linear factor of the form x - k. It's particularly useful for factoring polynomials and finding their roots. Here's how it works, based on the provided video reference.
Steps for Factoring Polynomials Using Synthetic Division
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Identify a Potential Root (k): This usually involves testing factors of the constant term of the polynomial. If you're given a factor to test, that gives you your k value (x-k=0, therefore x=k).
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Set up the Synthetic Division:
- Write the value of k (the potential root) outside of a division symbol.
- Write the coefficients of the polynomial to be divided inside the division symbol. Make sure to include a
0for any missing terms (e.g., if you have x³ + 2x + 1, you'd use the coefficients 1, 0, 2, 1).
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Perform the Synthetic Division:
- Bring down the first coefficient.
- Multiply the value you brought down by k, and write the result under the next coefficient.
- Add the two numbers in the column.
- Repeat the multiply and add steps until you reach the last coefficient.
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Interpret the Result:
- The last number in the bottom row is the remainder. If the remainder is 0, then k is a root of the polynomial, and (x - k) is a factor.
- The other numbers in the bottom row are the coefficients of the quotient polynomial, which is one degree lower than the original polynomial.
Example
Let's say you want to factor the polynomial 2x² + 5x - 6 + 9 and you suspect that x = 3 is a root (based on the video snippet).
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k = 3
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Setup:
3 | 2 5 -6 9 ------------------ -
Perform the Division:
3 | 2 5 -6 9 ------------------ 2- Bring down the 2.
3 | 2 5 -6 9 ------------------ 2 6- 3 times 2 is equal to 6.
3 | 2 5 -6 9 ------------------ 2 11- Add 5 and 6 and get 11.
The video snippet does not complete the division, however, the process continues in the same manner: multiply the divisor (3) by the most recent number in the result (11), place the result under the next number in the dividend (-6), and sum. Repeat this process as long as numbers remain in the dividend.
Factoring
If the remainder is 0, you've successfully found a factor. The quotient from the synthetic division provides the other factor.
For instance, if dividing x³ - x² - 14x + 24 by (x - 2) results in a remainder of 0 and a quotient of x² + x - 12, then:
x³ - x² - 14x + 24 = (x - 2)(x² + x - 12)
The quadratic x² + x - 12 can then be factored further into (x + 4)(x - 3).
Therefore, the fully factored form is:
x³ - x² - 14x + 24 = (x - 2)(x + 4)(x - 3)
Key Considerations
- Missing Terms: Remember to include placeholders (coefficients of 0) for any missing terms in the polynomial.
- Remainder: A zero remainder is crucial for confirming that the tested value is a root and (x - k) is a factor.
- Iterative Process: You may need to repeat synthetic division multiple times to fully factor a polynomial.
