Problem 55 - Algebra A Combined Function

Question

$$ \text { Write a polynomial function, } P(x) \text { , so that } P(0)=7 \text { . } $$

Step-by-Step Solution

Answer

Polynomial function is \( P(x) = 7 \).

Step 1: Understand the requirement

The problem asks us to find a polynomial function, denoted as \( P(x) \), which satisfies the condition \( P(0) = 7 \). This means that when we substitute \( x = 0 \) into the polynomial, the resulting value should be 7.

Step 2: Choose the simplest polynomial

To meet the condition \( P(0) = 7 \), we can start with the simplest polynomial, a constant polynomial. A constant polynomial is of the form \( P(x) = c \), where \( c \) is a constant value.

Step 3: Set the constant to meet the condition

Endeavour to satisfy the condition \( P(0) = 7 \). For a constant polynomial, \( P(x) = c \), when \( x = 0 \), the polynomial gives \( P(0) = c \). Therefore, we set \( c \) to 7 to meet the requirement.

Step 4: Write the polynomial function

The polynomial function that meets the condition is \( P(x) = 7 \). It's a constant function that outputs 7 for any value of \( x \), including \( x = 0 \). Therefore, \( P(0) = 7 \) as required.

Key Concepts

Constant Polynomial

A constant polynomial is a special type of polynomial where the degree is zero. It is termed "constant" because it does not involve any variable, such as \( x \). Hence, the polynomial function gives the same value irrespective of the input. Constant polynomials are expressed in the form \( P(x) = c \), where \( c \) is a constant. For example, \( P(x) = 7 \) is a constant polynomial as no matter what value of \( x \) you substitute, the result is always 7.
Understanding constant polynomials can be very helpful. They are simple but can be used to set or model specific conditions in algebraic problems. For instance, if we want a polynomial function to always output a certain number, like 7 in our example, then using a constant polynomial is the simplest way to achieve it.

Evaluating Polynomials

Evaluating polynomials involves finding the value of the polynomial expression for a particular value of \( x \). To evaluate a polynomial, you simply substitute the given number into the expression, replacing the variable \( x \). Let's break it down:

• Identify the polynomial function, like \( P(x) = 7 \).
• Substitute the value of \( x \) into the polynomial; for example, if evaluating at \( x = 0 \), you calculate \( P(0) \).
• Perform the arithmetic operation to find the result.

In our specific exercise, you only needed to ensure that when \( x = 0 \), \( P(x) = 7 \). Since \( P(x) = 7 \) is a constant polynomial, the evaluation is straightforward: \( P(0) = 7 \). This ability to substitute and evaluate at a point is a fundamental skill when working with functions in math.

Polynomial Notation

Polynomial notation is the way in which polynomial functions are expressed. Understanding this notation is important because it helps clearly communicate the polynomial and its properties. In general, a polynomial is written as \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_n, a_{n-1}, ..., a_1, \) and \( a_0 \) are coefficients, and \( n \) is the degree of the polynomial.
Key elements of polynomial notation include:

• The variable \( x \), which is raised to the power of integers.
• The coefficients (such as \( a_n \)) are the constants that multiply each term involving \( x \).
• The degree: the highest power of \( x \) that appears in the polynomial.

In the case of a constant polynomial like \( P(x) = 7 \), the notation simplifies significantly, as there are no variable terms, only the constant part. This makes it easy to understand and work with, especially when learning about polynomials.