One important concept in mathematics is finding the zeros of a function. By understanding how to find these zeros, you can solve various problems in mathematics, physics, and engineering.
In this article, we will discuss how to find all the zeros of the function f(x) = x^3 - 8x^2 - x + 8. This function is a cubic function, which is a polynomial of degree three.
To find the zeros of this function, we will use the factor by grouping method.
Firstly, we will group the terms by their similarity, which is the ratio of the first and second terms being the same as that of the third and fourth terms. This gives us:
x^3-8x^2-x+8 =(x^3-8x^2)-(x-8)
Next, we will factor out the greatest common factor from each of the groupings, which gives us:
=x^2(x-8)-1(x-8)
Now, we will use the distributive property to simplify this equation further. This gives us:
=(x^2-1)(x-8)
Finally, we will factor the quadratic expression x^2-1 into (x+1)(x-1). This gives us:
=(x-1)(x+1)(x-8)
Therefore, the zeros of the function f(x) = x^3 - 8x^2 - x + 8 are x = 1, x = -1, and x = 8.
In conclusion, finding the zeros of a function is an essential skill for solving various mathematical problems. By using the factor by grouping method, we can easily find the zeros of cubic functions like f(x) = x^3 - 8x^2 - x + 8. Remember to always practice and review mathematical concepts regularly to improve your understanding and proficiency.
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