Find the roots of the function (f(x) = x^3 - 9x^2 + 23x - 15) using the factor theorem.

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Accepted Answer

Step:1 ：The factor theorem states that a polynomial (f(x)) has a factor ((x - k)) if and only if (f(k) = 0). So we first need to find a value of (x) that makes (f(x) = 0).Step:2 ：By trying a few values, we find that (f(1) = 1^3 - 9 * 1^2 + 23 * 1 - 15 = 0), so ((x - 1)) is a factor of (f(x)).Step:3 ：Next, we can perform polynomial division to divide (f(x)) by ((x - 1)) to find the other factors. The result is (x^2 - 8x + 15), which can be factored into ((x - 3)(x - 5)).Step:4 ：Therefore, the factors of (f(x)) are ((x - 1)(x - 3)(x - 5)), and the roots of the function are (x = 1), (x = 3), and (x = 5).

Answer

Step: 1 ：The factor theorem states that a polynomial (f(x)) has a factor ((x - k)) if and only if (f(k) = 0). So we first need to find a value of (x) that makes (f(x) = 0).Step: 2 ：By trying a few values, we find that (f(1) = 1^3 - 9 * 1^2 + 23 * 1 - 15 = 0), so ((x - 1)) is a factor of (f(x)).Step: 3 ：Next, we can perform polynomial division to divide (f(x)) by ((x - 1)) to find the other factors. The result is (x^2 - 8x + 15), which can be factored into ((x - 3)(x - 5)).Step: 4 ：Therefore, the factors of (f(x)) are ((x - 1)(x - 3)(x - 5)), and the roots of the function are (x = 1), (x = 3), and (x = 5).

Find the roots of the function $f (x) = x^{3} - 9 x^{2} + 23 x - 15$ using the factor theorem.

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Popular Problems

Find the roots of the function f(x)=x3−9x2+23x−15 using the factor theorem.

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Popular Problems

Find the roots of the function $f (x) = x^{3} - 9 x^{2} + 23 x - 15$ using the factor theorem.