Problem 8 - Algebra and Trigonometry

Question

An expression is given. (a) Evaluate it at the given value. (b) Find its domain. $$ -x^{4}+x^{3}+9 x, \quad x=-1 $$

Step-by-Step Solution

Answer

(a) The value of the expression at \(x = -1\) is \(-11\). (b) The domain is all real numbers.

Step 1: Substitute the Value

To evaluate the expression at \(x = -1\), substitute \(-1\) for \(x\) in the expression \(-x^4 + x^3 + 9x\). This gives:\[-(-1)^4 + (-1)^3 + 9(-1)\]

Step 2: Simplify the Expression

Now, compute each term:- \(-(-1)^4 = -1\)- \((-1)^3 = -1\)- \(9(-1) = -9\)Thus, the expression simplifies to:\[-1 - 1 - 9\]

Step 3: Perform the Arithmetic Operations

Add the terms together:\[-1 - 1 - 9 = -11\]The value of the expression at \(x = -1\) is \(-11\).

Step 4: Determine the Domain of the Expression

For the polynomial \(-x^4 + x^3 + 9x\), there are no restrictions like division by zero or square roots of negative numbers. Therefore, the domain includes all real numbers.

Key Concepts

Evaluating Expressions

Evaluating an expression essentially means finding its value for a given input. In the problem provided, we have the expression \(-x^4 + x^3 + 9x\) which we need to evaluate at \(x = -1\). This involves plugging the value \(-1\) into the expression wherever you see the variable \(x\).
Here's how you do it step-by-step:

• Substitute \(-1\) for \(x\) in the equation: \(-x^4 + x^3 + 9x\) becomes \(-(-1)^4 + (-1)^3 + 9(-1)\).
• Calculate each term separately:
  - \(-(-1)^4 = -1\)
  - \((-1)^3 = -1\)
  - \(9(-1) = -9\).
• Combine these calculations to evaluate the expression as a whole:
  \(-1 - 1 - 9 \)
• Finally, sum the values: \(-1 - 1 - 9 = -11\).

Therefore, the value of the expression at \(x = -1\) is \(-11\). Evaluating expressions is an important skill in algebra, as it allows you to find specific values for algebraic expressions.

Domain of a Polynomial

The term 'domain' refers to all possible input values (often represented by \(x\)) that can be used in an expression. For a polynomial such as \(-x^4 + x^3 + 9x\), finding the domain is straightforward.
Unlike rational functions, which have variables in the denominator, or functions with square roots, polynomials do not have restrictions like division by zero or negative square roots.
Therefore, you don't need to worry about forbidden inputs! The domain of any polynomial, including the one in this problem, is always all real numbers. In mathematical terms, we express this as:

• The set of all real numbers: \(-\infty < x < \infty\).

Polynomials can be evaluated at any real number without any issues, making their domain quite extensive and simple to understand.

Real Numbers

Real numbers are one of the most fundamental concepts in mathematics. They encompass all the numbers you generally deal with in everyday life, including:

• Natural numbers like 1, 2, 3...
• Whole numbers which include zero along with natural numbers
• Integers like -3, -2, -1, 0, 1, 2, 3...
• Rational numbers which are fractions or ratios of two integers
• Irrational numbers like \( \pi \) and the square root of any non-perfect square.

In relation to addressing mathematical challenges like evaluating expressions or determining the domain of a polynomial, real numbers serve as the comprehensive field over which these considerations make sense. Real numbers provide a continuous spectrum that includes both rational and irrational numbers, making them incredibly versatile and complete for mathematical operations and solving problems.