Problem 4 - Algebra 2

Question

Find p(3) and p(-1) for each function. \(p(x)=x^{4}-3 x^{3}+2 x^{2}-5 x+1\)

Step-by-Step Solution

Answer

\(p(3) = 4\) and \(p(-1) = 12\).

Step 1: Substitute x = 3 into the polynomial

To find \( p(3) \), substitute \( x = 3 \) into the given polynomial \( p(x) = x^4 - 3x^3 + 2x^2 - 5x + 1 \). This results in the expression: \( 3^4 - 3 \times 3^3 + 2 \times 3^2 - 5 \times 3 + 1 \).

Step 2: Calculate each term with x = 3

First, calculate each term: \( 3^4 = 81 \), \( 3 \times 3^3 = 81 \), \( 2 \times 3^2 = 18 \), \( 5 \times 3 = 15 \). Substitute these values back into the expression: \( 81 - 81 + 18 - 15 + 1 \).

Step 3: Simplify the expression for p(3)

Simplify the expression: \( 81 - 81 = 0 \), then \( 18 - 15 = 3 \), and finally \( 3 + 1 = 4 \). Thus, \(p(3) = 4\).

Step 4: Substitute x = -1 into the polynomial

Now, find \( p(-1) \) by substituting \( x = -1 \) into the polynomial: \( (-1)^4 - 3(-1)^3 + 2(-1)^2 - 5(-1) + 1 \).

Step 5: Calculate each term with x = -1

Evaluate each term: \((-1)^4 = 1\), \(-3 \times (-1)^3 = 3\), \(2 \times (-1)^2 = 2\), \(-5 \times (-1) = 5\). Substitute back: \( 1 + 3 + 2 + 5 + 1 \).

Step 6: Simplify the expression for p(-1)

Add all terms: \(1 + 3 = 4\), then \(4 + 2 = 6\), then \(6 + 5 = 11\), and finally \(11 + 1 = 12\). Thus, \(p(-1) = 12\).

Key Concepts

Substitution Method

The substitution method is a powerful technique often used in evaluating polynomial functions. In this method, you replace a variable within an expression with a given value. This can help you quickly determine the value of the polynomial at specific points. For example, if you're asked to find \( p(3) \) for a polynomial function like \( p(x) = x^4 - 3x^3 + 2x^2 - 5x + 1 \), you would simply replace every instance of \( x \) with the number 3. Doing so transforms the expression into numerical operations you can perform step by step:

• \( 3^4 - 3 \times 3^3 + 2 \times 3^2 - 5 \times 3 + 1 \)
• This becomes \( 81 - 81 + 18 - 15 + 1 \)
• After simplifying, this equals 4.

The substitution method simplifies finding the output of complex expressions by turning them into basic arithmetic, making it an essential tool in evaluating polynomial functions.

Polynomial Function

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables. These functions are widely used in various fields of mathematics and science due to their flexibility and simplicity. A typical polynomial in a single variable looks like \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where:

• \(a_n, a_{n-1}, \ldots, a_1, a_0\) are coefficients
• \(n\) represents the highest power, or degree, of the polynomial
• \(x\) is the variable

Polynomials can be evaluated at specific points using the substitution method, which allows you to find the value of the function for specific values of \(x\). For example, the function \( p(x) = x^4 - 3x^3 + 2x^2 - 5x + 1 \) is a fourth-degree polynomial because the highest power of \(x\) is 4. Evaluating this polynomial for different values of \(x\) can help understand its behavior and solutions at various points.

Simplifying Expressions

Simplifying expressions is a process that involves performing operations to reduce an expression to its simplest form. This often involves combining like terms, performing arithmetic operations, and reducing the expression to a form that is easy to understand and evaluate.When evaluating polynomial functions through substitution, each substitution leads to a series of arithmetic operations which must then be simplified. For example:

• After substituting \( x = 3 \) into \( p(x) = x^4 - 3x^3 + 2x^2 - 5x + 1 \), the expression \( 81 - 81 + 18 - 15 + 1 \) can be simplified to 4.
• Similarly, substituting \( x = -1 \) results in \( 1 + 3 + 2 + 5 + 1 \), simplifying to 12.

Simplifying expressions not only makes it easier to arrive at results, but also makes the arithmetic more manageable and comprehensible. It's a fundamental skill in algebra that is essential when dealing with polynomial functions and other complex expressions.