Problem 10 - Algebra A Combined Function

Question

If \(P(x)=x^{3}+2 x-3\) and \(Q(x)=7 x+5,\) find each function value. $$ P(-1) $$

Step-by-Step Solution

Answer

The value of \( P(-1) \) is -6.

Step 1: Identify the Given Function

We are given the polynomial function \( P(x) = x^3 + 2x - 3 \). To find \( P(-1) \), we need to substitute \( -1 \) into the function in place of \( x \).

Step 2: Substitute -1 into the Function

Replace every \( x \) in \( P(x) = x^3 + 2x - 3 \) with \( -1 \): \[P(-1) = (-1)^3 + 2(-1) - 3\]

Step 3: Calculate the Cube of -1

Compute \((-1)^3\):\[(-1)^3 = -1\]So, the expression becomes:\[P(-1) = -1 + 2(-1) - 3\]

Step 4: Calculate the Multiplication

Calculate \(2 \times (-1)\): \[2(-1) = -2\]Now, the expression is:\[P(-1) = -1 - 2 - 3\]

Step 5: Simplify the Expression

Combine the terms: \[P(-1) = -1 - 2 - 3 = -6\]

Step 6: Conclusion

The function value at \( x = -1 \) is \( P(-1) = -6 \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving sums of powers of a variable. Each term in a polynomial has a coefficient and a non-negative integer exponent. Polynomials can include operations such as addition, subtraction, multiplication, and raising to a power. They are usually written in descending order of their exponents.

• Examples include linear functions ( x + c dot-dot dot dot dot dot dot3 dot dot dot dot dot dot dot2 dot2 dot dot dot dot dot dot dot3 dot dot dot dot dot dot3 dot3 ), quadratic functions ( x^2 + bx + c dot-dot-dot-dot-dot-dot-dot3 dot-dot-dot-dot-dot-dot-dot2 dot2 dot-dot-dot-dot-dot-dot-dotQ(x) = 7x + 5 dot), and higher-degree polynomials.

Polynomial functions have the following characteristics:

• The degree of the polynomial is determined by the highest exponent of the variable.
• The leading coefficient is the coefficient of the term with the highest degree.
• Polynomial functions are continuous and smooth, meaning they have no breaks, holes, or sharp corners.

Substitution Method

The substitution method is a straightforward technique used to evaluate functions for specific values of the variable. It involves replacing the variable in the function with a given numerical value. This helps in finding the function's value at that particular point.
To use the substitution method effectively:

• Identify the given function and the value to substitute.
• Replace every instance of the variable in the expression with the given number.
• Simplify the resulting expression to find the value of the function at the given point.

When using the substitution method in polynomial functions, it's important to carefully handle operations such as multiplication and exponentiation to ensure accuracy. This method is very useful, especially in polynomial arithmetic where functions need evaluation at specific points.

Polynomial Arithmetic

Polynomial arithmetic involves performing operations like addition, subtraction, multiplication, and division on polynomial functions. Understanding polynomial arithmetic is essential for simplifying and evaluating polynomials effectively.
Here are some key operations you might perform:

• Addition: Add polynomials by combining like terms, which are terms with the same variable raised to the same power.
• Subtraction: Subtract one polynomial from another by distributing a negative sign across the terms of the polynomial being subtracted and then combining like terms.
• Multiplication: Multiply each term in one polynomial by each term in the other, then combine like terms.
• Division: Divide polynomials using methods such as long division or synthetic division.

Understanding how to perform these operations allows for seamless manipulation and simplification of polynomials in problem-solving scenarios.

Cubic Functions

Cubic functions represent polynomial functions where the highest degree of the variable is three. These functions have the form \( ax^3 + bx^2 + cx + d \) where \(a, b, c,\) and \(d\) are constants. The presence of the \(x^3\) term gives cubic functions their distinct features.
Key characteristics of cubic functions include:

• Graph Shape: The graph of a cubic function can have a variety of shapes, often showing a curve with one or two bends. It can resemble an "S" shape if it crosses from one quadrant to another or showing a single inflection point.
• Number of Roots: A cubic function can have up to three real roots, where it intersects the \(x\)-axis.
• Turning Points: These functions can have two turning points, which are local minima or maxima.

Cubic functions play a key role in modeling real-world phenomena where relationships are not merely linear or quadratic, providing a richer framework for analysis and prediction.