Problem 71 - Elementary Algebra

Question

State the restrictions and then simplify. $$ 64-x 3 x 2-8 x+16 $$

Step-by-Step Solution

Answer

Simplified expression: \(-3x^2 - 8x + 80\); no restrictions.

Step 1: Identify the Expression

The given expression is \( 64 - 3x^2 - 8x + 16 \). We will break it down to simplify and identify any restrictions.

Step 2: Combine Like Terms

First, rewrite the expression grouping like terms: \( 64 + 16 - 3x^2 - 8x \), which simplifies to \( 80 - 3x^2 - 8x \).

Step 3: Simplify the Expression

Since there are no further like terms to combine, the simplified expression is \( -3x^2 - 8x + 80 \).

Step 4: Identify Any Restrictions

We look at the expression to find any restrictions. Since it is a polynomial, there are no denominators, radicals, or logarithms to restrict the domain. Thus, there are no restrictions for this simplification.

Key Concepts

Understanding Polynomials

Polynomials are algebraic expressions made up of terms that include variables raised to whole number exponents and coefficients. They can be classified based on the number of terms or the degree of the terms inside them. Each term in a polynomial is a combination of coefficients and variables. - **Monomial:** A single-term polynomial, like \(3x^2\). - **Binomial:** A two-term polynomial, such as \(x + 5\). - **Trinomial:** A three-term polynomial, for example, \(x^2 + x + 1\).The degree of a polynomial is determined by the term with the highest exponent. In the expression given, \(64 - 3x^2 - 8x + 16\), it simplifies over various steps to become \(-3x^2 - 8x + 80\). Here, it is a trinomial with a degree of 2 because the highest exponent of \(x\) is 2. Poly means 'many' and nomial means 'terms', making them key structures in algebra used to solve real-world problems.

Simplifying with Like Terms

'Like terms' are terms in an expression that have the same variable raised to the same power, but possibly different coefficients. Combining like terms is a fundamental step in simplifying polynomials. Imagine you are collecting apples; terms with the same variables are those apples. You cannot mix apples with oranges, hence, you can't combine different variable terms directly. For example:- In the expression \(64 - 3x^2 - 8x + 16\), the like terms \(64\) and \(16\) are constants, which add up to \(80\).- The terms \(-3x^2\) and \(-8x\) must be left as they are since they can’t be combined with each other or with the constants.Recognizing and combining like terms efficiently leads to a cleaner and simpler expression. It’s much like tidying up a closet by grouping similar items together, making further evaluations more streamlined.

Grasping Domain Restrictions

In algebra, domain restrictions are the values for which a function is undefined or cannot work. Polynomials, being expressions with positive whole number exponents, are very flexible, making them generally free of these restrictions under normal conditions.For functions, certain operations bring about domain restrictions:- **Denominators cannot be zero**: in fractions, any value that makes the denominator zero is excluded.- **Radicals with even roots must have non-negative radicands**: like square roots.- **Logarithms require positive arguments**.However, in the case of the given problem, \(64 - 3x^2 - 8x + 16\), which simplifies to \(-3x^2 - 8x + 80\), because this is a polynomial without any division, roots, or logarithms, there are no domain restrictions. This freedom is one reason polynomials are favored in calculations and modeling, providing solutions valid for all real numbers.