Questions: (4j^5 - 8j^4 + 3j^3 - 8j^2)^2 = (Simplify your answer. Type your answer in the form a + bj.)

Transcript text: \[ \left(4 j^{5}-8 j^{4}+3 j^{3}-8 j^{2}\right)^{2}= \] $\square$ (Simplify your answer. Type your answer in the form a + bj.)

Solution

Solution Steps

To simplify the given expression, we need to expand the square of the polynomial. This involves squaring each term and considering the cross-terms. However, this can be quite complex to do manually, so we will use Python to handle the algebraic expansion and simplification.

Step 1: Expand the Polynomial

We start with the polynomial $ P(j) = 4j^5 - 8j^4 + 3j^3 - 8j^2 $. To find $ (P(j))^2 $, we expand it as follows:

\[ (P(j))^2 = (4j^5 - 8j^4 + 3j^3 - 8j^2)^2 \]

Step 2: Calculate the Expanded Form

Upon expanding the polynomial, we obtain:

\[ (P(j))^2 = 16j^{10} - 64j^9 + 88j^8 - 112j^7 + 137j^6 - 48j^5 + 64j^4 \]

Step 3: Simplify the Expression

The expanded polynomial is already in its simplest form, expressed as a sum of powers of $ j $:

\[ 16j^{10} - 64j^9 + 88j^8 - 112j^7 + 137j^6 - 48j^5 + 64j^4 \]