Answered

Find the best solutions to your problems with the help of IDNLearn.com's expert users. Get prompt and accurate answers to your questions from our community of knowledgeable experts.

3. For the polynomial g(x) = x^4 - 8x^3 + 23^2 - 26x + 10, you are told that the zeros include x=1, x = -3 + i, and x = 3+i.
Explain how you know this list is incorrect. Then you are told that two of the given zeros are correct, with one of them having a multiplicity of 2. Use this information to determine all zeros of g(x). Be sure to show and explain all steps.

Sagot :

ChoiSungHyunidn

Step-by-step explanation:

If the polynomial has complex roots, the conjugates of those complex roots must also be part of the zeroes.

Therefore x = -3 - i and x = 3 - i are also zeroes of g(x). However this means that g(x) has 5 solutions when the degree of g(x) is 4. This is impossible, and hence the list is incorrect.

If either of the complex roots (x = -3 + i or x = 3 + i) had a multiplicity of 2, their conjugates will also have a multiplicity of 2, which is impossible as discussed previously.

Therefore x = 1 has a multiplicity of 2.

We have

x⁴ - 8x³ + 23x² - 26x + 10 = (x - 1)²(Ax² + Bx + C).

= (x² - 2x + 1)(Ax² + Bx + C)

= Ax⁴ + (B - 2A)x³ + (A + C - 2B)x² + (B - 2C)x + C

By Comparing Coefficients,

A = 1,

B - 2A = -8,

A + C - 2B = 23,

B - 2C = -26,

C = 10

Solving them we get A = 1, B = -6 and C = 10.

Hence x⁴ - 8x³ + 23x² - 26x + 10

= (x - 1)²(x² - 6x + 10).

Using the Quadratic Formula on x² - 6x + 10,

we find that the correct complex roots are

x = 3 + i and x = 3 - i.

Hence,

the zeroes of g(x) are x = 1, x = 3 + i and x = 3 - i.

Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.