Question

determine the total number of roots of each polynomial function using the factored form.

1. \\(f(x) = (x + 1)(x - 3)(x - 4)\\)

1. \\(f(x) = (x + 5)^3(x - 9)(x + 1)\\)

1. \\(f(x) = (x - 8)^2(x + 2)^2\\)

1. \\(f(x) = (x + 2)(x - 1)x - (4 + 3i)x - (4 - 3i)\\)

Explanation:

⚡ Using what you learned: Zeros and Multiplicity

Step 1: Identify the given polynomial function

The polynomial function in the bottom-right corner is:

$$ f(x) = (x + 2)(x - 1)[x - (4 + 3i)][x - (4 - 3i)] $$

Step 2: Count the linear factors

Each linear factor of the form \( (x - c)^k \) contributes \( k \) roots (counting multiplicity) to the total number of roots of the polynomial.

Let's list the factors and their exponents (multiplicities):

1. \( (x + 2) \) has an exponent of \( 1 \)
2. \( (x - 1) \) has an exponent of \( 1 \)
3. \( [x - (4 + 3i)] \) has an exponent of \( 1 \)
4. \( [x - (4 - 3i)] \) has an exponent of \( 1 \)

Step 3: Sum the multiplicities

Sum the exponents of all the factors to find the total number of roots:

$$ 1 + 1 + 1 + 1 = 4 $$

Answer:

4