Problem 12 - Algebra for College Students

Question

Determine whether each given value of x is a zero of the given function. See Example 1. $$x=5, \quad P(x)=x^{2}-5 x$$

Step-by-Step Solution

Answer

Yes, x = 5 is a zero of the function.

Step 1 - Substitute x in the function

Start by substituting the given value of x into the function. For x = 5, substitute it into the function: \[ P(x) = x^2 - 5x \] So, \[ P(5) = 5^2 - 5 \times 5 \]

Step 2 - Simplify the expression

Simplify the expression from step 1:\[ P(5) = 25 - 25 \]Combine like terms:\[ P(5) = 0 \]

Step 3 - Evaluate the result

Determine if the result from step 2 equals zero. Since \( P(5) = 0 \), this means that x = 5 is a zero of the function \( P(x) = x^2 - 5x \).

Key Concepts

Substitution

Substitution is a fundamental technique in algebra. It involves replacing a variable in a function with a given value. This can help us evaluate the function and check specific values.
Let's consider the function given in the exercise: \[P(x) = x^2 - 5x\] To determine if \(x = 5\) is a zero of this function, we substitute 5 for x. This means we'll replace every x in the function with 5:
\[P(5) = 5^2 - 5 \times 5\]
Substitution can help in various contexts, like solving equations or analyzing functions. Always make sure to substitute carefully to avoid mistakes.

Polynomial Functions

Polynomial functions involve sums of powers of a variable multiplied by coefficients. They have different degrees based on the highest power of the variable.
The function in this problem is a quadratic polynomial, which is of the form \(ax^2 + bx + c\). Here, \(P(x) = x^2 - 5x\), making it a simple quadratic function (since it has no constant term c).
Understanding polynomial functions involves recognizing their form and behavior. For instance:

• Linear functions (degree 1): \(ax + b\)
• Quadratic functions (degree 2): \(ax^2 + bx + c\)
• Cubic functions (degree 3): \(ax^3 + bx^2 + cx + d\)

Each type of polynomial function has its unique characteristics, like the number of roots, the shape of their graph, etc.

Simplifying Expressions

Simplifying expressions is crucial in mathematical problems to make them more manageable. It involves combining like terms and performing arithmetic operations.
In our function step by step solution, after substitution, we reached:
\[P(5) = 5^2 - 5 \times 5\]
We then simplify it by calculating each term:
\[5^2 = 25\] and \[5 \times 5 = 25\]
So,\[P(5) = 25 - 25 = 0\]
Simplifying helps in understanding the behavior and solving algebraic equations. Always remember to perform arithmetic accurately to ensure the correctness of the final result. If you simplify correctly, you will arrive at simplified results which are easier to interpret.