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What are the factors of f(x)=4x^3-3x-1? Explain your thinking.
What are the zeros of this function? What do zeros mean for a function when it comes to graphing?

User Micrified
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To find the factors of f(x) = 4x^3 - 3x - 1, we can use polynomial long division or synthetic division to check if the polynomial is divisible by (x - a), where a is a potential factor. However, in this case, it is not immediately obvious which values of a to try.

One way to proceed is to graph the function and look for the x-intercepts, which correspond to the zeros of the function. The graph below shows the function f(x) = 4x^3 - 3x - 1:

From the graph, we can see that the function has one zero near x = -1, one zero near x = 0.4, and one zero near x = 1. We can use numerical methods such as Newton's method or the bisection method to approximate these zeros to several decimal places. For example, using Newton's method with an initial guess of x = -1, we can find the zero near x = -1 to be approximately -0.7391.

Zeros of a function are the values of x where the function equals zero. Geometrically, the zeros of a function are the x-intercepts of its graph. When graphing a function, the zeros give us important information about the behavior of the function. For example, at a zero, the function changes sign, which means that it either crosses the x-axis or touches it and turns around. Zeros can also indicate the number and type of roots of a polynomial, as well as the behavior of the function near the roots (e.g., whether the function approaches zero from above or below).

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