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I_laff
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Why is it that for a 7th degree polynomial, the number of real roots is either 1, 3, 5, or 7?
There are a couple of reasons.I_laff said:Why is it that for a 7th degree polynomial, the number of real roots is either 1, 3, 5, or 7?
Let ##p(x)\in \mathbb{R}[x]## be our polynomial. If we had a root, say ##r_1##, then with the Euclidean algorithm, a long division, we get ##p(x)=q(x) \cdot (x-r_1)## since ##p(r_1)=0##. There is at least one real root, as the graph of ##p(x)## has to cross the ##x-##axis at least once, because it comes from ##+\infty## and goes to ##-\infty## or vice versa. Now we can go on with ##q(x)## which has degree ##6##. Either has ##q(x)## also a real root, which does not have to the case, or it has not. If it has, say ##r_2##, we continue the division by ##(x-r_2)##. But then we get a polynomial of degree ##5##, which thus again has to have a real root, because the degree is odd and the graph has again to cross the ##x-##axis. At the end, we will get an odd number of roots.I_laff said:Why is it that for a 7th degree polynomial, the number of real roots is either 1, 3, 5, or 7?
A polynomial degree is the highest exponent or power in a polynomial equation. It determines the number of solutions or roots that the equation can have.
The degree of a polynomial directly relates to the number of roots it can have. A polynomial of degree n can have at most n distinct roots. For example, a quadratic equation (degree 2) can have up to 2 roots, while a cubic equation (degree 3) can have up to 3 roots.
The degree of a polynomial determines the shape of its graph. For example, a polynomial of odd degree (1, 3, 5, etc.) will have a graph with both positive and negative values, while a polynomial of even degree (2, 4, 6, etc.) will have a graph that is either always positive or always negative. The degree also determines the number of turning points or local extrema in the graph.
No, a polynomial can have at most the same number of roots as its degree. This is because each root corresponds to a factor in the polynomial equation, and a polynomial of degree n can have at most n factors.
To find the roots of a polynomial, you can use various methods such as factoring, graphing, or using the quadratic formula. In some cases, the roots may be imaginary numbers. It is also important to note that not all polynomials can be easily solved for their roots.