SOLUTION: Use the given zero to find the remaining zeros of the polynomial function. (Enter your answers as a comma-separated list.) P(x) = x^3 + 11x^2 + x + 11; -i

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Question 1051320: Use the given zero to find the remaining zeros of the polynomial function. (Enter your answers as a comma-separated list.)
P(x) = x^3 + 11x^2 + x + 11; -i
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
Use the given zero to find the remaining zeros of the polynomial function. (Enter your answers as a comma-separated list.)
P(x) = x3 + 11x2 + x + 11; −i
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Since the coefficients of P(x) are rational, +i is also a zero.
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Therefore:: (x-i)(x+i) = x^2+1 is a factor of P(x)
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Divide P(x) by x^2+1 to get: x + 11
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So, P(x) = (x-i)(x+i)(x+11)
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Zeroes are :: -11, -i, i
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Cheers,
Stan H.
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Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


The first thing you need is that complex zeros always appear in conjugate pairs. So write the given zero in complex number, form: . The conjugate of is , so the companion zero to the one given is

If is a zero of a polynomial, then must be a factor of the polynomial. Hence, we now know two of the factors of the given polynomial, to wit:



and



Since this is a binomial conjugate pair, their product is the difference of two squares. Remember that . Hence:



Use polynomial long division to divide the original polynomial function by 

                                   x  +  11
              -----------------------------
x^2 + 0x + 1  |  x^3  +  11x^2  +  x  +  11
                 x^3      0x^2  +  x
                 --------------------------
                         11x^2  + 0x  +  11
                         11x^2  + 0x  +  11
                        -------------------
                                          0

So the third and final factor is and therefore the third and final zero is -11

By the way, saying x2 or x3 to mean x squared or x cubed is confusing. Use the caret mark (^) to indicate raising to a power, such as x^5 or e^x, which we all understand to mean or . If you want more information on rendering mathematical expressions in plain text, review Formatting Math as Text (Note that there are four pages of information).

John

My calculator said it, I believe it, that settles it
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