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Chapter 4: Q 30. (page 215)

In Problems 23鈥�30, use the given zero to find the remaining zeros of each function.

g(x)=2x5-3x4-5x3-15x2-207x+108, Zero:3i

Short Answer

Expert verified

The remaining zeros are卤3i,-12,3,4.

Step by step solution

01

Step 1. Given Information

The given polynomial is

g(x)=2x5-3x4-5x3-15x2-207x+108

Here we apply the conjugate the Pairs Theorem. The degree of the polynomial is 5.

3iis zero of polynomial then its conjugate -3ialso zero of the polynomial. Now we have two zeros
02

Step 2. Performing Long Division

Since all the coefficients in g(x)are real and 3iis a root of g(x)then -3iis also a root of g(x).

Then localid="1651210852073" (x+3i)(x-3i)are factors of gx. So by divides x-3ix+3i=x2+9

i.e. (x2+9)divides g(x)

Now, (2x5-3x4-5x3-15x2-207x+108)(x2+9)=2x3-3x2-23x+12

(Using polynomial division).

03

Step 3. Factors of the finding polynomial

Now,

2x3-3x2-23x+12=2x2(x-4)-5x(x-4)-3(x-4), by using zero method.

=(x-4)(2x2-5x-3)=(x-4)(2x2-6x+x-3)=(x-4){2x(x-3)+1(x-3)}=(x-4)(2x+1)(x-3)

Then the zeros are 卤3i,-12,3,4.

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