Problem 22 - Beginning and Intermediate Algebra

Question

Evaluate each polynomial when a \(k=2\) and b) \(k=-3\) $$3 k^{3}-10 k-11$$

Step-by-Step Solution

Answer

a) When \(k=2\), the value of the polynomial is \(-7\). b) When \(k=-3\), the value of the polynomial is \(-62\).

Step 1: Substitute the value of \(k\)

For this part, we substitute \(k=2\) into the given polynomial: \(3(2)^3 - 10(2) - 11\).

Step 2: Simplify

Now we'll simplify our expression by following the order of operations (PEMDAS): Exponents, then Multiplication, and finally Addition and Subtraction. \[ \begin{aligned} 3(2)^3 - 10(2) - 11 &= 3(8) - 20 - 11 \\ &= 24 - 20 - 11 \\ &= 4 - 11 \\ &= -7 \end{aligned} \] So, when \(k=2\), the value of the polynomial is \(-7\). b) When \(k=-3\):

Step 1: Substitute the value of \(k\)

Now we substitute \(k=-3\) into the given polynomial: \(3(-3)^3 - 10(-3) - 11\).

Step 2: Simplify

Again, we'll simplify our expression by following the order of operations (PEMDAS): \[ \begin{aligned} 3(-3)^3 - 10(-3) - 11 &= 3(-27) + 30 - 11 \\ &= -81 + 30 - 11 \\ &= -51 - 11 \\ &= -62 \end{aligned} \] So, when \(k=-3\), the value of the polynomial is \(-62\). In summary, we have evaluated the polynomial for two different values of \(k\): a) the value of the polynomial when \(k=2\) is \(-7\) and b) when \(k=-3\) is \(-62\).

Key Concepts

Substitution Method

The substitution method is a powerful and straightforward tool used in algebra to evaluate expressions or solve equations. It involves replacing variables with given numerical values to simplify the expressions further. In the context of polynomial evaluation, this technique helps us calculate the polynomial's value for specific values of the given variables.

To apply the substitution method, follow these steps:

• Identify the variable in the given polynomial or equation.
• Replace the variable with the provided value, making sure to enclose it in parentheses if necessary.
• Proceed with simplifying the expression using basic arithmetic operations.

This method is handy for quickly assessing how a polynomial behaves when different numbers are substituted for its variables.

Order of Operations

In mathematics, expressions can have multiple arithmetic operations, and it is crucial to perform them in a specific order to ensure accurate results. The order of operations, often remembered by the acronym PEMDAS, is a rule used to clarify which procedures should be performed first.

Let's break down the components:

• P: Parentheses - solve any operations inside parentheses first.
• E: Exponents - resolve any exponentiation next.
• M/D: Multiplication and Division - perform these operations from left to right.
• A/S: Addition and Subtraction - lastly, handle these from left to right.

For our given polynomial, adhering to PEMDAS ensures that we evaluate exponents and multiplication prior to any subtraction operations.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and arithmetic operations. They are the key components in algebra and allow us to express mathematical relationships in a generalized form. Understanding them is critical for setting up and solving problems effectively.

Components of algebraic expressions include:

• Variable: Symbols like \( k \) that represent unknown values or quantities.
• Constant: Numerical values that do not change, such as -11 in the polynomial.
• Operators: Signs like +, -, *, and / that indicate mathematical operations.

In polynomial expressions, the variables are typically raised to different powers, influencing the expression's complexity and overall behavior.

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. The base is multiplied by itself as many times as the exponent indicates. It is a fundamental component in mathematics, especially in algebra, as it allows for concise representation of repeated multiplication.

In our example, we have:\( 3k^3 \). Here:

• Base: \( k \)
• Exponent: 3, which tells us to multiply \( k \) by itself three times.

Exponents make expressions more compact and are essential for representing higher degree polynomials efficiently. Understanding how exponentiation works is crucial for evaluating more complex polynomials accurately.