Problem 67 - Elementary and Intermediate Algebra

Question

Evaluate each polynomial for \(a=-2\) and \(b=3 .\) See Example 4. $$ 5 a b^{3}-a b-b+10 $$

Step-by-Step Solution

Answer

The polynomial evaluates to -257.

Step 1: Identify the expression

The polynomial that needs to be evaluated is given by \(5 a b^{3} - a b - b + 10\). We will substitute the values \(a = -2\) and \(b = 3\) into this polynomial.

Step 2: Substitute the variables

Replace \(a\) with \(-2\) and \(b\) with \(3\) in the expression:\[5(-2)(3)^{3} - (-2)(3) - 3 + 10\]

Step 3: Calculate the powers

Evaluate \(b^3\) which is \(3^3 = 27\).

Step 4: Simplify step-by-step

Now the expression becomes:\[5(-2)(27) + 2(3) - 3 + 10\]Start by calculating each term individually. Calculate:1. \(5(-2)(27) = -270\)2. \(2(3) = 6\)Thus the expression simplifies to:\[-270 + 6 - 3 + 10\]

Step 5: Finish the calculation

Perform the addition and subtraction:1. Combine \(-270 + 6 = -264\)2. Combine \(-264 - 3 = -267\)3. Combine \(-267 + 10 = -257\)

Step 6: Conclusion

The value of the polynomial \(5 a b^{3} - a b - b + 10\) evaluated at \(a = -2\) and \(b = 3\) is \(-257\).

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical expressions made up of variables, coefficients, and operations such as addition, subtraction, and multiplication. These expressions can contain one or more terms, and each term is a product of a constant coefficient and variables raised to a non-negative integer power.

For example, the polynomial expression given in the exercise is:

• \[5ab^{3} - ab - b + 10\]

• It consists of four distinct terms.
• The first term, \(5ab^3\), includes both variables \(a\) and \(b\) raised to the third power.
• The second and third terms are \(-ab\) and \(-b\), showing subtraction and the use of variable \(b\).
• The last term is a constant \(10\), showing the interplay between different types of terms in a polynomial expression.

Understanding how these components work together is crucial. Polynomials are foundational in algebra as they can model a wide variety of real-world situations and form the basis for more complex mathematical concepts.

Substitution in Polynomials

Substitution in polynomials involves replacing the variables in a polynomial expression with specific numerical values. This method allows us to calculate the value of the expression for these particular values of the variables.

In this exercise, we're given specific values for \(a\) and \(b\):

• \(a = -2\)
• \(b = 3\)

The substitution is carried out by replacing every occurrence of \(a\) in the polynomial with \(-2\) and every occurrence of \(b\) with \(3\).

After substitution, the expression

• \[5(-2)(3)^3 - (-2)(3) - 3 + 10\]

Your job is to follow this correctly as any small misstep can lead to a wrong answer.

Substitution is not just a task of plug-and-chug. It shows the dynamic nature of polynomials, where the same expression can yield different values based on the variables given. This makes substitution a powerful tool in algebra, simplifying problems and leading us toward solutions.

Algebraic Simplification

Algebraic simplification involves breaking down expressions to their simplest form by executing all possible calculations. This process is crucial when working with polynomials to make expressions more manageable and to find their values quickly and efficiently.

In the given exercise, after substituting the values for \(a\) and \(b\), we have:\[5(-2)(27) + 2(3) - 3 + 10\]The simplification proceeds by evaluating each term individually.

• First, calculate the value of each term independently, such as \(-270\) from \(5(-2)(27)\) and \(6\) from \(2(3)\), ensuring each step is accurate.
• Finally, combine all simplified terms together through addition and subtraction:
  • \(-270 + 6 = -264\)
  • \(-264 - 3 = -267\)
  • \(-267 + 10 = -257\), ensuring each operation maintains the integrity of the expression.

Simplifying expressions not only makes them easier to work with but is also essential in verifying correct answers. Every algebraic operation impacts the outcome, thus careful execution and rechecking of steps are important in simplifying expressions successfully.