Problem 14 - Algebra Form and Function

Question

Is the expression a polynomial in the given variable? $$ (x-1)(x-2)(x-3)(x-4)+29, \text { in } x $$

Step-by-Step Solution

Answer

Given expression: \((x-1)(x-2)(x-3)(x-4) + 29\) Simplified expression: \(x^4 - 10x^3 + 31x^2 - 40x + 53\) Answer: Yes, the given expression is a polynomial in the variable x.

Step 1: Multiplying the binomials

In order to simplify the expression, we must multiply the binomials \((x-1)(x-2)(x-3)(x-4)\). We will first multiply the first two binomials, and the last two binomials separately. \((x-1)(x-2)\) and \((x-3)(x-4)\). \((x-1)(x-2) = x^2 - 3x + 2\) \((x-3)(x-4) = x^2 - 7x + 12\)

Step 2: Multiply the two simplified expressions

Now, we need to multiply \((x^2 - 3x + 2)\) and \((x^2 - 7x + 12)\). \((x^2 - 3x + 2)(x^2 - 7x + 12) = x^4 - 10x^3 + 31x^2 - 40x + 24\)

Step 3: Add the constant term

Finally, add the constant term 29 to the simplified expression. \(x^4 - 10x^3 + 31x^2 - 40x + 24 + 29 = x^4 - 10x^3 + 31x^2 - 40x + 53\)

Step 4: Check if the expression is a polynomial

Now that we have simplified the given expression, let's check if it meets the definition of a polynomial. The simplified expression is: \(x^4 - 10x^3 + 31x^2 - 40x + 53\) This is an algebraic expression that is the sum of a finite number of monomials (terms) involving the variable \(x\), with non-negative integer exponents. Therefore, the given expression is a polynomial in the variable \(x\).

Key Concepts

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. Algebraic expressions can consist of single terms or multiple terms connected by these operations.
For example, in the expression \((x-1)(x-2)(x-3)(x-4) + 29\), we see a combination of binomials being multiplied together and a constant being added. When working with algebraic expressions, it's important to recognize the basic parts:

• **Variables**: Symbols like \(x\) that represent unknown numbers.
• **Constants**: Numbers like 1, 2, 3, and 29 that have fixed values.
• **Operators**: Signs like \(+, -, \times, \div\) which indicate mathematical operations.

Algebraic expressions do not include equality signs—those appear in equations, which show equality between two expressions. Here, we are specifically looking to identify the polynomial nature within the algebraic expression.

Binomials

Binomials are algebraic expressions that contain exactly two terms. These take the general form \(a + b\) or \(a - b\), where \(a\) and \(b\) can be numbers, variables, or a combination of both. In our problem, expressions such as \((x-1)\) and \((x-2)\) are examples of binomials. When multiplying binomials, each term in the first binomial is multiplied by each term in the second binomial. This process can be visualized as:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outer terms in each product.
• Inner: Multiply the inner terms in each product.
• Last: Multiply the last terms in each binomial.

This multiplication process is often abbreviated as FOIL (First, Outer, Inner, Last), especially for multiplying two binomials. Understanding how to handle binomials is crucial for simplifying more complex algebraic expressions.

Monomials

Monomials are algebraic expressions with only one term. A monomial can be a number, a variable, or a product of numbers and variables. Examples include \(3x^2\), \(50\), or \(xy\). In polynomial expressions, each individual term separated by addition or subtraction signs is considered a monomial. For instance, in a polynomial like \(x^4 - 10x^3 + 31x^2 - 40x + 53\), each term is a separate monomial. Monomials generally take the form \(a \cdot x^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer that represents the power of the variable. The degree of a monomial is the sum of the exponents of all its variables. When dealing with polynomials, identifying and understanding each monomial is essential. It helps in determining properties such as degree, leading coefficient, and behavior of the polynomial as a whole. In our example, the expression consists of several monomials like \(x^4\) and \(-40x\), which, when combined, form the polynomial.