Problem 40 - Beginning and Intermediate Algebra

Question

Let \(f(x)=3 x-7\) and \(g(x)=x^{2}-4 x-9 .\) Find each of the following and simplify. $$g(p-5)$$

Step-by-Step Solution

Answer

The simplified expression for \(g(p-5)\) is \(p^2 - 14p + 36\).

Step 1: Substituting x with (p-5) in g(x)

Given the function \(g(x) = x^2 - 4x - 9\), let's substitute \(x\) with \((p - 5)\): \(g(p-5) = (p-5)^2 - 4(p-5) - 9\)

Step 2: Expanding (p-5)^2

To expand \((p-5)^2\), we use the formula \((a - b)^2 = a^2 - 2ab+ b^2\). Here, \(a = p\) and \(b = 5\): \((p-5)^2 = p^2 - 2\cdot p\cdot 5 + 5^2\) Now, substitute this back in the expression for \(g(p-5)\): \(g(p-5) = (p^2 - 10p + 25) - 4(p-5) - 9\)

Step 3: Distributing -4 to (p-5)

Now, we will distribute the -4 to the terms inside the parentheses: \(-4(p-5) = -4p + 20\) Substitute this back into the expression for \(g(p-5)\): \(g(p-5) = (p^2 - 10p + 25) - 4p + 20 - 9\)

Step 4: Simplifying the expression

Combine the like terms in the expression: \(g(p-5) = p^2 - 10p + 25 - 4p + 20 - 9\) \(g(p-5) = p^2 - 14p + 36\) Thus, the simplified expression for \(g(p-5)\) is \(p^2 - 14p + 36\).

Key Concepts

Substitution

Substitution in mathematics refers to the process of replacing a variable in an expression or equation with a specific value or another expression. This technique is widely used to simplify complex equations or to evaluate functions at specific points.
For the function evaluation of \(g(x) = x^2 - 4x - 9\), we are asked to substitute \(x\) with \(p-5\). By performing this substitution, the expression \(g(p-5)\) becomes \((p-5)^2 - 4(p-5) - 9\).
This changed expression represents the function value at a new input, \(p-5\), instead of \(x\), enabling us to further work on simplifying or analyzing it.

Expand Binomial

Expanding a binomial involves applying algebraic identities to express a binomial raised to a power as a sum or difference of terms. A common formula for expanding a binomial squared is \((a - b)^2 = a^2 - 2ab + b^2\).
In our example, we need to expand \((p-5)^2\). This process steps through as follows:

• The term \(a\) in the formula is \(p\), and \(b\) is \(5\).
• Substitute into the identity: \((p-5)^2 = p^2 - 2(p)(5) + 5^2\).
• Simplifying further gives us \(p^2 - 10p + 25\).

This expanded form is then used to continue the simplification of the function \(g(p-5)\). Expanding binomials is essential for simplifying polynomial expressions, allowing us to work with individual terms.

Simplify Algebraic Expression

Simplifying an algebraic expression involves combining like terms to make the expression easier to understand or work with. This process reduces the expression to its simplest form, frequently used to make further calculations more straightforward or solve equations.
After expanding and substituting into the function \(g(p-5) = p^2 - 10p + 25 - 4(p-5) - 9\), the next step involves combining like terms. Begin by distributing \(-4\) across \(p-5\) to get \(-4p + 20\).
Then, place all pieces in the expression:

• Combine \(-10p\) and \(-4p\) to result in \(-14p\).
• Add the constants \(25\), \(20\), and \(-9\) to get \(36\).

Thus, the simplified expression is \(p^2 - 14p + 36\). Properly simplifying polynomial expressions through combining like terms helps in understanding the structure and form of algebraic equations.

Polynomial Functions

Polynomial functions are a type of mathematical function that involves variables raised to whole number exponents and coefficients. They often appear in the form \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\), where \(a_n\), \(a_{n-1}\), ..., and \(a_0\) are constants, and \(n\) is a non-negative integer.
The function \(g(x) = x^2 - 4x - 9\) is a polynomial function of degree 2, also known as a quadratic function. When we substitute \(p-5\) into this function and simplify, it remains a polynomial, specifically \(g(p-5) = p^2 - 14p + 36\).
Key features of polynomial functions include:

• They are continuous and smooth graphs.
• The degree of the polynomial determines the shape and number of turning points in the graph.
• They can be added, subtracted, multiplied, and divided (except by another polynomial of degree 0).

Understanding polynomial functions and being able to manipulate them is crucial in algebra as they form the foundation for more complex mathematical concepts.