Describe a method you can use to shorten the list of possible rational zeros when using the rational zero theorem? I know how to do the problems, I just don't know how to shorten that method. Please help me.
at wikipedia,
http://en.wikipedia.org/wiki/Rational_root_theorem
about 2/3 of the way down the page there's a discussion on shortening the list of rational roots, by substituting in x=(r+t) for x where r is not one of the roots.
To shorten the list of possible rational zeros when using the rational zero theorem, you can follow these steps:
1. Write down all the possible rational zeros. These are the ratios of the factors of the constant term (the term without any variables) to the factors of the leading coefficient (the coefficient of the highest degree term).
2. Simplify the fractions as much as possible. For example, if one of the possible zeros is 2/4, simplify it to 1/2.
3. Use the synthetic division method or long division to test each possible zero. Start with the one that seems most likely.
4. If the synthetic division or long division produces a remainder of zero, the tested value is indeed a root or zero of the polynomial. Otherwise, it is not a root.
5. If a tested value is a root, divide the polynomial by the linear factor obtained from the synthetic division or long division. This will give you a new polynomial, which will have a lower degree than the original one.
By following these steps, you can eliminate possible zeros that do not produce a remainder of zero. This helps to shorten the list of possible rational zeros and narrows down your search for the actual roots of the polynomial.
To shorten the process of finding possible rational zeros using the rational zero theorem, you can follow these steps:
1. Write down the equation in its standard form, with all terms on one side equal to zero.
2. Identify the leading coefficient and the constant term. The leading coefficient is the coefficient of the highest degree term, and the constant term is the constant on its own.
3. List all the factors of the constant term. For example, if the constant term is 12, the factors would be 1, 2, 3, 4, 6, and 12.
4. List all the factors of the leading coefficient. For example, if the leading coefficient is 5, the factors would be 1 and 5.
5. Apply the rational zero theorem, which states that the possible rational zeros are all the ratios with a numerator from the factors of the constant term and a denominator from the factors of the leading coefficient. This means you divide each factor of the constant term by each factor of the leading coefficient. For example, if the constant term factors are ±1, ±2, ±3, ±4, ±6, and ±12, and the leading coefficient factors are ±1 and ±5, your list of possible rational zeros would include:
±1/1, ±1/5, ±2/1, ±2/5, ±3/1, ±3/5, ±4/1, ±4/5, ±6/1, ±6/5, ±12/1, and ±12/5.
6. Simplify the fractions if possible. For example, ±2/1 simplifies to ±2.
7. Cross off any duplicate or equivalent zeros from your list.
By following these steps, you can efficiently shorten the list of possible rational zeros using the rational zero theorem.