# Write a polynomial equation with rational coefficients

### Write a polynomial equation with rational coefficients

Again it is important to note that multiple roots of the same value should be counted separately. Key Terms sign: positive or negative polarity. Thus, there are either two or zero positive roots for this polynomial. However, it does not tell the exact number of positive or negative roots. Sections: Finding quadratics from their zeroes , Finding polynomials from a list of values How about if they give you more than just one or two roots? It follows from the fundamental theorem of algebra and a fact called the complex conjugate root theorem, that every polynomial with real coefficients can be factorized into linear polynomials and quadratic polynomials without real roots. Learning Objectives Use the rule of signs to find out the maximum number of positive and negative roots a polynomial has Key Takeaways Key Points The rule of signs gives us an upper bound number of positive or negative roots of a polynomial. The advantage of this is that once we have found a root, we immediately have found the smaller degree polynomial of which we again wish to find the roots and the rational root theorem will provide us with even fewer candidates for this root. Moreover, once we have established a root, we must use division anyway to check whether it is a multiple root. But the process for finding polynomials from their zeroes works the same as for finding quadratics from their zeroes. This number, or any number less than it by a multiple of 2, could be the number of positive roots. You know that any nice neat whole-number or fractional root turns into a nice neat linear factor. And any messy root with a square root or a complex number in it turns into a paired root.

Therefore, we know that it has at most two negative roots. Then, we must count the number of sign differences between consecutive nonzero coefficients. Change the exponents of the odd-powered coefficients, remembering to change the sign of the first term. It is used in the proof of the fundamental theorem of arithmetic. Then the procedure is the same; count the number of sign changes between consecutive nonzero coefficients.

The disadvantage is that we have to use long division more often. One way to find zeros of a polynomial is trial and error.

### Find a third degree polynomial with zeros

This number, or any number less than it by a multiple of 2, could be the number of positive roots. Using this knowledge, you take the roots they give you, create all of your factors, multiply them together, and then either you tack on the "I don't know if you divided something out" constant a, or else, if they give you a point from the line, you use that point to solve for a. Change the exponents of the odd-powered coefficients, remembering to change the sign of the first term. This is why factorization is so important: to be able to recognize the zeros of a polynomial quickly. The disadvantage is that we have to use long division more often. A more efficient way is through the use of the Rational Zero Theorem. Thus if you have found such a factorization of a given function, you can be completely sure what the zeros of that function are. It is important to note that for polynomials with multiple roots of the same value, each of these roots is counted separately. Once we have found all the rational zeros and counted their multiplicity, for example, by dividing using long division , we know the number of irrational and complex roots.

When given a polynomial with integer coefficients, we can plug in all of these candidates and see whether they are a zero of the given polynomial.

The Rule of Signs The rule of signs gives an upper bound number of positive or negative roots of a polynomial. Then your answer will be a polynomial of degree higher than 2. The rule states that if the terms of a polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less by a multiple of 2.

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