Problem 21 Evaluate each polynomial when a ... [FREE SOLUTION]

Chapter 6: Problem 21

Evaluate each polynomial when a $k=2$ and b) $k=-3$ $$k^{2}+5 k+8$$

Short Answer

Expert verified

When $k=2$, the evaluated polynomial is $22$, and when $k=-3$, the evaluated polynomial is $2$.

Step by step solution

Substitute the variable k

We will first evaluate the polynomial when $k=2$ and then when $k=-3$. So, we substitute the values of $k$ into the equation: For $k=2$ $$ (2)^{2}+5\cdot2+8 $$ For $k=-3$ $$ (-3)^{2}+5\cdot(-3)+8 $$

Simplify the polynomial after substituting the values

Now, we will simplify the polynomial for each value of $k$. For $k=2$ $$ (2)^{2}+5\cdot2+8 = 4+10+8 = 22 $$ For $k=-3$ $$ (-3)^{2}+5\cdot(-3)+8 = 9-15+8 = 2 $$ So, the evaluated polynomial for $k=2$ is $22$, and for $k=-3$ is $2$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method

The substitution method is a powerful tool used in algebra to evaluate or solve equations by replacing variables with their given numerical values. In the context of polynomial evaluation, this method involves taking each variable in the polynomial and substituting it with the specified value provided in the question. This is often the first step in the process of evaluating a polynomial.Let's break it down:

Identify the variable that needs to be substituted. In our exercise, the variable is $k$.
Replace $k$ with the given values, one at a time. For example, first use $k=2$ and then $k=-3$.

By substituting the given values into the polynomial equation, we convert an algebraic expression into a numerical one, which can then be simplified further.

Simplifying Polynomials

Simplifying polynomials involves the process of combining like terms and performing arithmetic operations to reduce the expression to its simplest form. After substituting the values in place of the variables, the next critical step is to simplify the expression to find the result.Consider the steps involved:

Perform any exponentiation, such as squaring the substituted value of $k$. For instance, $ (2)^2 $ becomes $4$ and $(-3)^2$ becomes $9$.
Next, carry out multiplication. Multiply the coefficient by the substituted variable value, such as $5\cdot2 = 10$ or $5\cdot(-3) = -15$.
Lastly, sum up all the terms in the expression, including constants, to reach the final result.

Through simplification, a potentially complex polynomial expression can be reduced to a simple numeric answer, allowing for easier interpretation and understanding.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols but does not have an equality sign like an equation does. Polynomials, such as the one in our exercise, are a specific type of algebraic expression consisting of variables raised to whole-number exponents and multiplied by coefficients.Key components of polynomial algebraic expressions:

Variables: Symbols such as $k$ that represent unknown quantities and can be replaced with different values during evaluation.
Constants: Numbers without variables, like $8$ in the expression, that remain unchanged regardless of the variable's value.
Terms: The components of the polynomial separated by addition or subtraction signs. In this case, $k^2$, $5k$, and $8$ are separate terms.

Understanding algebraic expressions is vital for solving mathematical problems, as it allows for the manipulation and evaluation of equations based on mathematical rules and operations, ultimately leading to finding solutions like we did in this evaluation process.

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91影视

Evaluate each polynomial when a \(k=2\) and b) \(k=-3\) $$k^{2}+5 k+8$$

Short Answer

Step by step solution

Substitute the variable k

Simplify the polynomial after substituting the values

Key Concepts

Substitution Method

Simplifying Polynomials

Algebraic Expressions

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