Problem 68 - Elementary and Intermediate Algebra

Question

Evaluate each polynomial for \(a=-2\) and \(b=3 .\) See Example 4. $$ -a^{3} b+a b-a-21 $$

Step-by-Step Solution

Answer

The value of the polynomial is \(-1\).

Step 1: Substitute the variables

Begin by substituting the given values of the variables into the polynomial: replace every occurrence of \(a\) with \(-2\) and every occurrence of \(b\) with \(3\). The polynomial becomes:\(-(-2)^3 (3) + (-2)(3) - (-2) - 21.\)

Step 2: Evaluate the powers and products

First, calculate \((-2)^3\), which is \(-8\). Then multiply this result by \(3\), yielding \(-24\). Note the negative sign in front: \(-(-24)\) becomes \(+24\). Next, calculate \((-2)\times(3)\), which is \(-6\). Now substitute these values back into the polynomial expression:\(24 - 6 + 2 - 21.\)

Step 3: Perform the remaining operations

Add and subtract the values obtained from the previous step:1. Add \(24 + 2\) to get \(26\).2. Subtract \(6\) from \(26\) to get \(20\).3. Finally, subtract \(21\) from \(20\) to get \(-1\).

Key Concepts

Variable Substitution

In mathematics, variable substitution is a fundamental technique used to simplify expressions or solve equations. It involves replacing a variable with a given value, which helps transform an algebraic expression into a numerical one. This is especially useful when evaluating polynomials where specific values are provided for the variables.

Let's consider the polynomial given in the exercise: \[-a^{3} b+a b-a-21\]. Instead of dealing with abstract variables, substitution allows us to plug in the actual values. Here, we substitute \(a = -2\) and \(b = 3\). This changes the polynomial to a simpler form:

• Replace \(a\) with \(-2\)
• Replace \(b\) with \(3\)

By doing this, the expression becomes:\[-(-2)^3 (3) + (-2)(3) - (-2) - 21.\]

This process not only simplifies the polynomial but also sets the stage for further calculations, ultimately leading to a concrete numerical solution.

Evaluating Polynomials

Evaluating polynomials involves replacing variables with numbers and then simplifying the result. It's an essential skill when trying to determine the output of a polynomial given specific inputs.
To evaluate the polynomial \[-(-2)^3(3) + (-2)(3) - (-2) - 21\], we start by dealing with the exponents and then proceed with multiplication, addition, and subtraction operations.
First, calculate the cube of \(-2\):

• \((-2)^3 = -8\)
• Multiply by \(3\), resulting in \(-24\).

The expression becomes \[-(-24)\], which simplifies to \(+24\) as the negative sign in front changes the sign.
Next, calculate \((-2) \times 3 = -6\), so the polynomial simplifies to \[24 - 6 + 2 - 21.\]

This set of operations shows how each step systematically simplifies the polynomial until you find the final result. This careful arithmetic breakdown is crucial for arriving at the correct solution.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations. They can be simple, like a single variable, or complex, involving multiple terms with variables raised to different powers. Understanding these expressions is foundational in algebra as it helps us manipulate and solve more complicated problems.

The polynomial we are working with, \[-a^{3} b+a b-a-21\], is an algebraic expression with terms derived from multiplications and additions of variables \(a\) and \(b\). Each term of the polynomial represents a component of the expression:

• \(-a^3b\): a product of a cubic term and another variable.
• \(+ab\): a product of two variables.
• \(-a\): a standalone term with one variable.
• \(-21\): a constant term with no variables.

Substituting values for these variables and simplifying allow us to "solve" the polynomial to find what it equals when specific values are used. Algebraic expressions are the building blocks of these calculations, and knowing how to work with them is essential in both basic and advanced algebra.