91Ó°ÊÓ

When evaluating polynomials, the substitution method is a crucial technique. The process involves replacing the variables in the polynomial with the given values. This step is essential because it transforms an abstract expression into concrete numbers, making it easier to evaluate.

Let's consider the given exercise: Evaluate the polynomial \(5 + 40x - \frac{1}{2}x^2 \) when \(x = 0\).

Step 1: Identify the polynomial and the given value. We have \(5 + 40x - \frac{1}{2}x^2\) and \(x = 0\).
Step 2: Substitute the value of \(x\) into the polynomial. This means we replace every \(x\) in the polynomial with \(0\): \(5 + 40(0) - \frac{1}{2}(0)^2\).

By following this method, we change the polynomial into a simpler form, ready for the next step, which is simplification.

After substituting the given value into the polynomial, the next step is simplification. Simplifying expressions involves performing arithmetic operations to reduce the polynomial to its simplest form.

For our exercise, after substitution, we have the expression \(5 + 40(0) - \frac{1}{2}(0)^2\).

Let's break it down:

• First, calculate \(40(0)\), which equals \(0\).
• Next, address the term \( - \frac{1}{2}(0)^2\). Since \(0^2 = 0\), \(- \frac{1}{2}(0)^2 = 0\) as well.

Now our expression looks like this: \(5 + 0 - 0\).

Finally, add the terms together. \(5 + 0 - 0 = 5\). Through these simplification steps, we see that the polynomial reduces to the single number \(5\).

Evaluating polynomials involves finding the polynomial's value for specific variable inputs. It's a fundamental concept in algebra that helps understand the behavior of polynomial functions.

In our exercise, we evaluated the polynomial \(5 + 40x - \frac{1}{2}x^2 \) at \(x = 0\).

• First, we used substitution to replace \(x\) with \(0\).
• Then, through simplification, we calculated each term's value and combined them.

The final result of the polynomial evaluation is \(5\). By following these systematic steps – substitution and simplification – we can efficiently evaluate any polynomial for given variable values. This methodical approach helps solve polynomial problems not just in school exercises, but also in real-world applications.