Problem 55 - Elementary Algebra

Question

The expression \(\frac{4 x^{3}}{9 x-7}\) is not a polynomial because

Step-by-Step Solution

Answer

Answer: The expression \(\frac{4x^3}{9x-7}\) is not a polynomial because it contains a variable in the denominator with a negative exponent. A polynomial is defined as an algebraic expression with variables, coefficients, and non-negative whole number exponents. In this case, the expression can be rewritten as \(4x^3(9x-7)^{-1}\), which contains a term with a negative exponent, thus not fitting the definition of a polynomial.

Step 1: Defining a Polynomial

A polynomial is an algebraic expression made up of variables, coefficients, and exponents, where exponents are non-negative whole numbers (integers greater than or equal to 0). A polynomial has the general form: \[P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\] Where \(a_n, a_{n-1},..., a_1, a_0\) are constants, and \(n\) is a non-negative whole number.

Step 2: Identifying the Key Differences

In the given expression, \(\frac{4x^3}{9x-7}\), the numerator is a polynomial, \(4x^3\), as it meets the definition mentioned above. However, the denominator is also an algebraic expression, \(9x - 7\). This fraction doesn't fit the definition of a polynomial as it contains a variable in the denominator.

Step 3: Explaining Why a Polynomial Cannot Have Fractions

A polynomial cannot have fractions because of the requirement that exponents must be non-negative whole numbers. If there is a fraction in the polynomial, it means there is a variable in the denominator with a negative exponent. In the case of the given expression, it can be rewritten as: \[\frac{4x^3}{9x-7} = 4x^3(9x-7)^{-1}\] As we can see, now there is a term with a negative exponent, which doesn't fit the definition of a polynomial. Therefore, the given expression is not a polynomial.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can represent a wide range of mathematical scenarios. Typically, they don't have an equals sign unless they're part of an equation.

An algebraic expression can be as simple as a number or consist of complex combinations:

• Simple: Just a constant or a single variable like x or 7.
• Complex: Can include multiple terms, such as 3x + 2y - 5.

In a fraction, the numerator or denominator can each be algebraic expressions. However, for a whole expression to be called a polynomial, it must not involve any variables in the denominator.

Exponents

Exponents represent repeated multiplication of the same number. In the expression x³, the exponent 3 tells us to multiply x by itself three times, or x · x · x.

Key characteristics of exponents in polynomials include:

• Non-negative integers: All exponents in polynomials must be whole numbers (0, 1, 2,...).
• Indicate power: They express how many times the variable is used in multiplication.

When a polynomial expression is turned into a fraction, like in \(\frac{4x^3}{9x-7}\), it might introduce negative exponents upon manipulation, which violate polynomial rules.

Variables and Coefficients

Variables are symbols used to represent unknown or changeable values, like x or y in algebraic expressions. They serve as placeholders that can take on many values.

Coefficients are the numbers in front of these variables, indicating their multiplicative factor.

• In the term 4x³, 4 is the coefficient, and x is the variable.
• Each term may consist of a coefficient multiplied by a variable raised to a power.

When forming polynomials, every term follows this structure without any terms having negative coefficients or exponents in the polynomial context.