Problem 61 - Elementary and Intermediate Algebra

Question

Evaluate each polynomial for \(a=-2\) and \(b=3 .\) See Example 4. $$ 6 a^{2} b $$

Step-by-Step Solution

Answer

The value of the polynomial is 72.

Step 1: Understand the given expression

The expression given is a polynomial: \(6a^2b\). Here, \(a\) and \(b\) are variables, and we are asked to substitute the values \(a = -2\) and \(b = 3\).

Step 2: Substitute the values into the polynomial

Replace \(a\) with \(-2\) and \(b\) with \(3\) in the expression \(6a^2b\). This gives us:\(6(-2)^2(3)\).

Step 3: Evaluate the expression

First, evaluate \((-2)^2\) which is \((-2)\times(-2) = 4\). Replace \((-2)^2\) in the expression:\(6 imes 4 imes 3\).

Step 4: Calculate the result

Multiply the numbers together:\(6 imes 4 = 24\),then, multiply \(24\) by \(3\):\(24 imes 3 = 72\).

Key Concepts

Understanding Substitution in Polynomials

Substitution is a crucial concept in algebra, especially when dealing with polynomials. By rewriting an expression with specific values for different variables, you can simplify and solve complex algebraic equations.
To practice substitution, you'll often replace variables with given numbers. In the exercise at hand, a polynomial like \(6a^2b\) is presented, where you have to substitute \(a = -2\) and \(b = 3\). This means you literally swap these numbers in place of the variables in the expression.
Consider substitution as a way of "testing" the polynomial with specific values, transforming abstract expressions into concrete numbers you can calculate. As you master this, your ability to evaluate different polynomials accurately will greatly improve.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They're essentially mathematical phrases that represent quantities and relationships. An example is the polynomial \(6a^2b\), where numeric coefficients (6) multiply a variable \(a\) raised to a power and another variable \(b\).
Understanding the structure of algebraic expressions is crucial for correct evaluation. The order matters for both variables and operations. For instance, in \(6a^2b\), \(a^2\) indicates \(a\) is squared before being multiplied by numbers or other variables.
Every component of an algebraic expression plays a part in shaping the overall outcome when values are substituted, highlighting how each part interconnects in mathematical relationships.

The Benefit of Step-by-Step Solutions

Approaching problems via a step-by-step solution is like following a well-charted map. It organizes thought processes and ensures no details are overlooked.
In this exercise, each step, from understanding the expression to calculating the final result, assures clarity. First, comprehend the polynomial structure. Next, substitute given values. Then, evaluate powers and products in stages, showing growth from one step to the next.
This method reinforces foundational concepts, focusing on accurate execution of algebraic manipulation at each stage. Through practice and repetition, you'll gain confidence and skill in tackling more complex mathematical challenges. This structured approach enhances precision and understanding in algebraic problem-solving.