Problem 13 - Algebra for College Students

Question

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=1, \quad P(x)=x^{3}-x^{2}+x-1$$

Step-by-Step Solution

Answer

Yes, x = 1 is a zero of the function.

Step 1 - Substitute the value of x

Substitute the given value of x into the polynomial function. Here the value is x = 1, so substitute 1 into the function: P(1) = 1^3 - 1^2 + 1 - 1

Step 2 - Simplify the expression

Simplify the expression obtained by substituting x = 1. Calculate each term separately: 1^3 = 1, -1^2 = -1, 1, -1. Therefore, P(1) = 1 - 1 + 1 - 1.

Step 3 - Combine like terms

Combine the simplified terms to find the value of P(1): 1 - 1 + 1 - 1 = 0.

Step 4 - Determine if it is a zero

Since P(1) = 0, the given value x = 1 is indeed a zero of the function P(x).

Key Concepts

Substitute Value in Polynomial

Substituting a value in a polynomial is the first step to determine if it is a zero of the polynomial function. Start by taking the given value of x and replace every instance of x in the polynomial with this value.

For example, if the polynomial function is P(x) = x^3 - x^2 + x - 1 and the value to check is x = 1, every occurrence of x in P(x) should be replaced by 1. This gives us:
P(1) = 1^3 - 1^2 + 1 - 1.

Accurately substituting values into each term ensures the proper evaluation of the polynomial at the specified value.

Simplify Polynomial Expression

After substituting the value, the next step is to simplify the polynomial expression. To do this, evaluate each substituted term independently.

For the example P(1) raised before:
Calculate 1^3 = 1, 1^2 = 1, and replace accordingly in the polynomial: 1 - 1 + 1 - 1.

Careful simplification ensures that the terms are computed correctly, leading to accurate results.

Combine Like Terms

Combining like terms involves adding or subtracting the polynomial terms that are similar. This step is essential in further simplifying the polynomial expression.

For the given example:
P(1) = 1 - 1 + 1 - 1.
Combine the constant terms together to get: 0.

Doing this helps reduce the polynomial to a simpler form, making it easy to evaluate.

Determine Zeros of Polynomial

To determine if the substituted value is a zero of the polynomial, check the final simplified expression. If the result is 0, then the given value is indeed a zero of the polynomial.

In our example:
After simplifying and combining like terms, we have P(1) = 0.
Since the result is 0, x = 1 is verified as a zero of P(x).

Determining zeros is crucial for understanding the roots of the polynomial and solving polynomial equations.