Chapter 2.5, Problem 53E

a.

To determine

To write the factored form of the polynomial function of degree 4 whose zeros are $-1,2,i$ .

Also, $f(1)=8$ .

Answer to Problem 53E

  $f(x)=-2(x+1)(x-2)(x-i)(x+i)$

Explanation of Solution

Given:

Zeros of polynomial function: $-1,2,i$

Value: $f(1)=8$

Concept used:

Linear Factorization Theorem:

If $f(x)$ is a polynomial of degree n, where n > 0, then f has precisely a linear factors

  $f(x)=a(x-c_1)(x-c_2)\mathrm{........}(x-c_n)$

Calculation:

As $i$ is a zero of the polynomial, the conjugate of $i$ is also a zero of the polynomial.

Conjugate of $i$ is $-i$ .

So, the zeros are $-1,2,i,-i$

Using above concept, $f(x)$ can be written as,

  $\begin{array}{l}\Rightarrow f(x)=a(x-(-1))(x-(2))(x-(i))(x-(-i))\\ \Rightarrow f(x)=a(x+1)(x-2)(x-i)(x+i)\end{array}$

According to question, $f(1)=8,$

  $\begin{array}{l}\Rightarrow f(1)=8\\ \Rightarrow a(1+1)(1-2)(1-i)(1+i)=8\\ \Rightarrow a(2)(-1)(1+i-i-i^2)=8\\ \Rightarrow a(2)(-1)(1-(-1))=8\\ \Rightarrow a(2)(-1)(1+1)=8\\ \Rightarrow a(2)(-1)(2)=8\\ \Rightarrow a(-4)=8\\ \Rightarrow a=-2\end{array}$

So, polynomial function can be written as $f(x)=-2(x+1)(x-2)(x-i)(x+i)$ .

Conclusion:

Therefore, the polynomial function in factored form is $f(x)=-2(x+1)(x-2)(x-i)(x+i).$

b.

To determine

To write the polynomial form of the polynomial function of degree 4 whose zeros are $-1,2,i$ .

Answer to Problem 53E

  $f(x)=-2x^4+2x^3+2x^2+2x+4$

Explanation of Solution

Calculation:

From above answer, the factored form of polynomial function is

  $f(x)=-2(x+1)(x-2)(x-i)(x+i).$

Now, writing the above form as polynomial form,

  $\begin{array}{l}\Rightarrow f(x)=-2(x+1)(x-2)(x-i)(x+i)\\ \Rightarrow f(x)=-2(x^2-2x+x-2)(x^2+xi-xi-i^2)\\ \Rightarrow f(x)=-2(x^2-x-2)(x^2-(-1))[\because i^2=-1]\\ \Rightarrow f(x)=-2(x^2-x-2)(x^2+1)\\ \Rightarrow f(x)=-2(x^4+x^2-x^3-x-2x^2-2)\\ \Rightarrow f(x)=-2(x^4-x^3-x^2-x-2)\\ \Rightarrow f(x)=-2x^4+2x^3+2x^2+2x+4\end{array}$

Conclusion:

Therefore, the polynomial function in polynomial form is $f(x)=-2x^4+2x^3+2x^2+2x+4.$