Let's go through each part of the question and solve it.

Question 1: Polynomial Evaluation Using Synthetic Division and Remainder Theorem

Given:

$f(x) = 2x^5 + 5x^4 - 4x^2 + 3x - 1$

We need to find:

1. $f(2)$
2. $f\left(\frac{1}{2}\right)$

a) Find $f(2)$ using Synthetic Division

To find $f(2)$ using synthetic division:

1. Set up the synthetic division table using $2$ as the divisor.
2. The coefficients of $f(x)$ are: $[2, 5, 0, -4, 3, -1]$ (Note: There’s no $x^3$ term, so we include a $0$ for that term).

Using synthetic division, the final remainder will be $f(2)$.

Let's go through these calculations.

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Related Questions

1. How does synthetic division work for polynomials of different degrees?
2. Why is the Remainder Theorem useful for evaluating functions?
3. What if we used direct substitution instead of synthetic division?
4. How does the initial velocity affect the height in the second problem?
5. How would the time to hit the ground change if the initial height was different?

Tip

For synthetic division, always ensure to include zero coefficients for any missing terms in the polynomial.