Question

determine the total number of roots of each polynomial function.

1. (f(x) = 3x^5 + 2x^4 + x^3 - 8x^2)
2. (f(x) = (3x^2 + 1)^2)
3. (g(x) = 5x - 12x^2 + 3)
4. (g(x) = (x - 5)^2 + 2x^5)

Explanation:

🆕 New Concept Discovered: Fundamental Theorem of Algebra
The highest exponent of a polynomial determines its total number of complex roots.

Step 1: Analyze the first function

The first function is:
\[ f(x) = 3x^5 + 2x^4 + x^3 - 2x^2 \]

The highest exponent (degree) of this polynomial is \( 5 \). By the Fundamental Theorem of Algebra, the degree of a polynomial equals its total number of complex roots (including real and imaginary roots, counted with multiplicity).

Total roots = \( 5 \)

Step 2: Analyze the second function

The second function is:
\[ g(x) = 5x - 10x^2 + 3 \]

Rearranging the terms in descending order of power:
\[ g(x) = -10x^2 + 5x + 3 \]

The highest exponent (degree) is \( 2 \).

Total roots = \( 2 \)

Step 3: Analyze the third function

The third function is:
\[ f(x) = (3x^4 + 1)^2 \]

To find the degree, we look at the term with the highest power when expanded:
\[ (3x^4)^2 = 9x^8 \]

The highest exponent (degree) is \( 8 \).

Total roots = \( 8 \)

Step 4: Analyze the fourth function

The fourth function is:
\[ g(x) = (x - 5)^3 + 2x^5 \]

When expanding \( (x - 5)^3 \), the highest power term is \( x^3 \).
Comparing this with the other term in the function, \( 2x^5 \), the overall highest exponent (degree) of the polynomial is \( 5 \).

Total roots = \( 5 \)

Answer:

• For \( f(x) = 3x^5 + 2x^4 + x^3 - 2x^2 \): 5
• For \( g(x) = 5x - 10x^2 + 3 \): 2
• For \( f(x) = (3x^4 + 1)^2 \): 8
• For \( g(x) = (x - 5)^3 + 2x^5 \): 5