QuestionJuly 15, 2025

Find all of the zeros of the function shown. f(x)=x^4-6x^3+10x^2-6x+9 x=-9,i,-i x=3,i,-i x=3,3i,-3i x=9,i,-i x=-3,i,-i x=3,3i,-3i

Find all of the zeros of the function shown. f(x)=x^4-6x^3+10x^2-6x+9 x=-9,i,-i x=3,i,-i x=3,3i,-3i x=9,i,-i x=-3,i,-i x=3,3i,-3i
Find all of the zeros of the function shown. f(x)=x^4-6x^3+10x^2-6x+9 x=-9,i,-i x=3,i,-i x=3,3i,-3i x=9,i,-i x=-3,i,-i x=3,3i,-3i

Solution
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Answer

x=3, \frac{3+i\sqrt{3}}{2}, \frac{3-i\sqrt{3}}{2} Explanation 1. Identify possible rational roots Use the Rational Root Theorem. Possible rational roots are \pm 1, \pm 3, \pm 9. 2. Test x=3 as a root Substitute x=3 into f(x). Calculate f(3) = 3^4 - 6 \cdot 3^3 + 10 \cdot 3^2 - 6 \cdot 3 + 9 = 0. Thus, x=3 is a root. 3. Perform synthetic division by (x-3) Divide f(x) by (x-3) to find the quotient polynomial. Result: x^3 + 0x^2 + 0x + 3. 4. Factor the quotient polynomial The quotient x^3 + 3 can be factored as (x+3)(x^2 - 3x + 3). 5. Solve x^2 - 3x + 3 = 0 Use the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} with a=1, b=-3, c=3. Calculate discriminant: (-3)^2 - 4 \cdot 1 \cdot 3 = -3. Roots are x = \frac{3 \pm i\sqrt{3}}{2}. 6. List all zeros Combine results: x=3, x=\frac{3+i\sqrt{3}}{2}, x=\frac{3-i\sqrt{3}}{2}.

Explanation

1. Identify possible rational roots<br /> Use the Rational Root Theorem. Possible rational roots are $\pm 1, \pm 3, \pm 9$.<br /><br />2. Test $x=3$ as a root<br /> Substitute $x=3$ into $f(x)$. Calculate $f(3) = 3^4 - 6 \cdot 3^3 + 10 \cdot 3^2 - 6 \cdot 3 + 9 = 0$. Thus, $x=3$ is a root.<br /><br />3. Perform synthetic division by $(x-3)$<br /> Divide $f(x)$ by $(x-3)$ to find the quotient polynomial. Result: $x^3 + 0x^2 + 0x + 3$.<br /><br />4. Factor the quotient polynomial<br /> The quotient $x^3 + 3$ can be factored as $(x+3)(x^2 - 3x + 3)$.<br /><br />5. Solve $x^2 - 3x + 3 = 0$<br /> Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=-3$, $c=3$. <br /> Calculate discriminant: $(-3)^2 - 4 \cdot 1 \cdot 3 = -3$.<br /> Roots are $x = \frac{3 \pm i\sqrt{3}}{2}$.<br /><br />6. List all zeros<br /> Combine results: $x=3$, $x=\frac{3+i\sqrt{3}}{2}$, $x=\frac{3-i\sqrt{3}}{2}$.
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