( \( f(x)=7 x^{4}-6 x^{3}+174 x^{2}-150 x-25 \) a) Find the rational zeros of \( f(x) \), and then the other zeros. b) Factor \( f(x) \) into linear factors. a) Find the rational zeros of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The rational zeros are \( 1,-\frac{1}{7} \). (Use a comma to separate answers as needed. Express complex numbers in terms of \( i \).) B. There are no rational zeros. Find the other zeros of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The other zeros are (Use a comma to separate answers as needed. Express complex numbers in terms of \( i \).) B. There are no other zeros.
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Finding the rational zeros of polynomials is quite a journey! In this case, you can apply the Rational Root Theorem. By testing values, it turns out that \( f(1) = 0 \) making \( 1 \) a rational zero. So, the rational zeros of \( f(x) \) are \( 1 \) and \( -\frac{1}{7} \). For the other zeros, after factoring \( f(x) \) using synthetic division, it helps to use the quadratic formula on the resulting polynomial. You will find the other zeros to include complex numbers as well! Now, let’s talk about factoring! The polynomial \( f(x) \) can be rewritten using its rational zeros, giving you the expression \( (x - 1)\left(7x^{3} + x^{2} + 165x + 25\right) \). Next, you can perform polynomial division to factor that cubic further. Ultimately it becomes \( f(x) = (x - 1)\left(7x + 5\right)(x + \frac{1}{7})(x^2 + 165) \), allowing you to break it down into linear factors. Happy factoring!
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