Problem 54 - Algebra Form and Function

Question

Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) $$ h(1) $$

Step-by-Step Solution

Answer

Question: Find the value of \(h(1)\), where \(h(x) = f(x)g(x)\), given the functions \(f(x) = 2x^3 + 3x - 3\) and \(g(x) = 3x^2 - 2x - 4\). Answer: -6

Step 1: Multiply the functions \(f(x)\) and \(g(x)\)

To find the product of \(f(x) = 2x^3 + 3x - 3\) and \(g(x) = 3x^2 - 2x - 4\), multiply each term in \(f(x)\) by every term in \(g(x)\) and then add the results together: \(h(x)= (2x^3)(3x^2) + (2x^3)(-2x) + (2x^3)(-4) + (3x)(3x^2) +(3x)(-2x) +(3x)(-4) -3(3x^2) -3(-2x) -3(-4)\)

Step 2: Simplify the expression

Next, perform the indicated operations and combine the like terms to get the simplified expression for \(h(x)\): \(h(x)= 6x^5 - 4x^4 - 8x^3 + 9x^3 - 6x^2 - 12x - 9x^2 +6x + 12\) \(h(x)= 6x^5 - 4x^4 + x^3 - 15x^2 + 6\)

Step 3: Find the value of \(h(1)\)

Now, let's find the value of \(h(1)\) by plugging in \(x = 1\) to the expression of \(h(x)\): \(h(1) = 6(1)^5 - 4(1)^4 + (1)^3 - 15(1)^2 + 6\) \(h(1) = 6 - 4 + 1 - 15 + 6 = \boxed{-6}\)

Key Concepts

function evaluation

Function evaluation is a crucial concept when working with algebraic expressions. It involves substituting a specific value for the variable in a function and then simplifying the expression to find its result. In the problem provided, the task was to evaluate the function \( h(x) \) at \( x = 1 \). This is done by replacing every instance of \( x \) in the polynomial expression with the number 1.
For example:

• In \( h(x) = 6x^5 - 4x^4 + x^3 - 15x^2 + 6 \), we substitute \( x = 1 \).
• Then calculate: \( 6(1)^5 - 4(1)^4 + 1 - 15(1)^2 + 6 \).
• The result, after simplification, is: \( h(1) = -6 \).

Function evaluation helps to determine the value of a polynomial at specific points, which can be useful in graphing or identifying characteristics of the function, such as zeros and intercepts.

polynomial simplification

Polynomial simplification involves reducing a polynomial expression to its simplest form. After multiplying two polynomials such as \( f(x) = 2x^3 + 3x - 3 \) and \( g(x) = 3x^2 - 2x - 4 \), the result is a more complex polynomial.
The next step is to combine like terms to simplify it.

• For example, terms like \( 9x^3 \) and \(-8x^3 \) are combined into a single term \( x^3 \).
• This reduction process continues until all like terms are combined, simplifying the expression.

By simplifying the polynomial, you transform the expression into a more manageable and understandable form, making it easier to evaluate and use further in functions or equations.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operators (such as +, −, ×, ÷). They form the building blocks of many algebra problems, including polynomial expressions.
In the given exercise, we work with complex algebraic expressions characterized by multiple terms of varying degrees. Each term consists of a variable raised to a power, known as the degree, and a coefficient, the numerical factor multiplying the variable.

• For instance, in the term \( 2x^3 \), the degree is 3 and the coefficient is 2.
• Managing these expressions involves understanding how to manipulate these components through operations like addition, subtraction, multiplication, and simplification.

Mastery of algebraic expressions is key for solving equations, understanding functions, and comprehending higher-level mathematical concepts. It allows algebra students to transition from working with basic numbers to complex variable expressions used in various scientific and engineering fields.