Question

The product, \(P(n),\) of two numbers is represented by the expression \(2 n^{2}-4 n+3,\) where \(n\) is a real number. a) If one of the numbers is represented by \(n-3,\) what expression represents the other number? b) What are the two numbers if \(n=1 ?\)

Short answer

The other number is \(2n + 2\). When \(n = 1\), the numbers are \(-2\) and \(4\).

Step-by-step solution

Understand the given product

Given the product of two numbers as an expression: \(P(n) = 2n^2 - 4n + 3\).

Identify one of the numbers

It's given that one of the numbers is \(n - 3\).

Set up the equation for the other number

Let the other number be represented by \(A\). Then, the product of the two numbers can be expressed as: \((n - 3) \cdot A = 2n^2 - 4n + 3\).

Solve for the second number

To find the second number, we can isolate \(A\): \[A = \frac{2n^2 - 4n + 3}{n - 3}\].

Perform polynomial division

Divide \(2n^2 - 4n + 3\) by \(n - 3\) using polynomial division to get: \[A = 2n + 2\].

Substitute \(n = 1\) to find the numbers

To find the numbers when \(n = 1\), substitute 1 into the expressions \(n - 3\) and \(2n + 2\). \[ (1 - 3) = -2\]\[ 2(1) + 2 = 4\].

Key concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that involve variables and coefficients. These expressions are composed of terms, which are variables raised to a power, multiplied by coefficients. For example, in the given expression, the polynomial is represented as: \[2n^2 - 4n + 3\]. Here:

• The term \(2n^2\) has a coefficient of 2 and the variable \(n\) raised to the power of 2.
• The term \(-4n\) has a coefficient of -4 and the variable \(n\) raised to the power of 1.
• The term 3 is a constant, which means it doesn’t involve any variables.

Polynomial expressions can be added, subtracted, multiplied, and divided. Understanding these operations is essential for solving problems involving polynomials.

Equations

An equation is a mathematical statement that asserts the equality of two expressions. In this problem, we deal with the equation derived from the given product: \[(n - 3) \cdot A = 2n^2 - 4n + 3\]. Here, the left side represents the product of two numbers, and the right side is the polynomial representation of that product. To find the unknown factor 'A,' we need to rearrange the equation and isolate 'A'. Through polynomial division, we can solve for \(A\)and express it in terms of \(n\). This gives us:\[A = \frac{2n^2 - 4n + 3}{n - 3}\]. Dividing polynomials is similar to long division, but it involves matching and reducing like terms, leading us to the simplified expression for the other factor.

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes both rational and irrational numbers. In our problem, \(n\) is specified as a real number. Real numbers are significant in algebra because they provide a complete and continuous set for building mathematical models. For instance, when \(n = 1\), and we substitute 1directly into the expressions derived from the product, we calculate real, specific outcomes. Plugging \(n = 1\)into \(n - 3\) and \(2n + 2\)gives us two numbers: [-2, 4].These solutions are real numbers, ensuring the values make logical sense within the context of the problem.