Problem 51 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. $$g(r+4)$$

Step-by-Step Solution

Answer

The short answer is: \(g(r+4) = r^2 + 15r + 46\).

Step 1: Identify the given function

The given function is: \[g(x) = x^2 + 7x + 2\]

Step 2: Substitute x with (r+4)

Substitute \(x\) with \(r+4\) in the given function: \[g(r+4) = (r + 4)^2 + 7(r + 4) + 2\]

Step 3: Expand (r+4)^2 and simplify

First, expand \((r+4)^2\), which is \((r+4)(r+4)\): \[(r+4)^2 = r^2 + 8r + 16\] Now substitute the expanded expression back into the function: \[g(r+4) = (r^2 + 8r + 16) + 7(r + 4) + 2\]

Step 4: Distribute and combine like terms

Now distribute the constant in the second term and combine like terms: \[g(r+4) = r^2 + 8r + 16 + 7r + 28 + 2\] Combine the like terms: \[g(r+4) = r^2 + (8r + 7r) + (16 + 28 + 2)\]

Step 5: Simplify the expression

Simplify the expression by combining the constants and coefficients: \[g(r+4) = r^2 + 15r + 46\] The simplified expression for \(g(r+4)\) is \(r^2 + 15r + 46\).

Key Concepts

Understanding Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, connected by addition, subtraction, and multiplication. These expressions can include terms with varying powers of a variable. For example, in the function \( g(x) = x^2 + 7x + 2 \), you'll notice:

• A quadratic term: \( x^2 \)
• A linear term: \( 7x \)
• A constant term: \( 2 \)

Each part plays a specific role, dictated by the power of the variable. The degree of a polynomial is determined by the highest exponent present—in this case, 2, making it a quadratic polynomial. Understanding polynomials helps in breaking down and efficiently solving algebraic problems.

Algebraic Substitution Simplified

Algebraic substitution is a powerful technique where you replace a variable with another expression to simplify or solve an equation. In our exercise, we substituted \( x \) with \( r + 4 \) in the given polynomial function \( g(x) \). This produces a new function: \[ g(r+4) = (r + 4)^2 + 7(r + 4) + 2 \] This process involves simple replacement and allows you to evaluate the function for a specific value or expression. By substituting, complex expressions can become manageable, enabling further simplification and evaluating of equations more easily.

Exploring Binomial Expansion

Binomial expansion is a critical concept used to expand expressions that are raised to a power. In our specific problem, we expand \((r+4)^2\). To expand: - Recognize this as \((r + 4)(r + 4)\).- Distribute each term: \[ (r + 4)(r + 4) = r^2 + 4r + 4r + 16 \] - Combine like terms to get: \[ r^2 + 8r + 16 \] Binomial expansion is helpful for simplifying polynomials and getting them ready for solving or graphing. Understanding how to work with binomials makes it easier to handle a wide range of algebraic expressions.