Question: owling polynomial. \[ m(x) = 3x^4 - 13x^3 + x^2 + 13x - 4 \] e polynomial division and the…

owling polynomial.

\[ m(x) = 3x^4 - 13x^3 + x^2 + 13x - 4 \]

e polynomial division and the quadratic formula, if necessary, to identify the actual zeros.

e answers with commas.

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Solution

Find the zeros of the polynomial function: \[ m(x) = 3x^{4} - 13x^{3} + x^{2} + 13x - 4 \] Apply the Rational Root Theorem to identify possible rational zeros. Possible rational zeros are: \[ \pm1, \pm2, \pm4, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{4}{3} \] Test \( x = 1 \): \[ m(1) = 3(1)^4 - 13(1)^3 + 1^2 + 13(1) - 4 = 0 \] Thus, \( x = 1 \) is a root. Perform polynomial division by \( (x - 1) \): \[ m(x) = (x - 1)(3x^{3} - 10x^{2} - 9x + 4) \] Now, find the roots of \( 3x^{3} - 10x^{2} - 9x + 4 \). Possible rational zeros are the same as before. Test \( x = 4 \): \[ 3(4)^3 - 10(4)^2 - 9(4) + 4 = 0 \] Thus, \( x = 4 \) is a root. Perform polynomial division by \( (x - 4) \): \[ 3x^{3} - 10x^{2} - 9x + 4 = (x - 4)(3x^{2} + 2x - 1) \] Now, solve \( 3x^{2} + 2x - 1 = 0 \) using the quadratic formula: \[ x = \frac{-2 \pm \sqrt{4 + 12}}{6} \] \[ x = \frac{-2 \pm \sqrt{16}}{6} \] \[ x = \frac{-2 \pm 4}{6} \] Thus, \[ x = \frac{1}{3}, \quad x = -1 \] Therefore, the zeros of the polynomial are: \[ x = 1, \quad x = 4, \quad x = \frac{1}{3}, \quad x = -1 \]