Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (x-1)(x+1) $$

Short answer

The product is \( x^2 - 1 \).

Step-by-step solution

Identify the special product formula

Recognize that \( (x-1)(x+1) \) is a difference of squares. The formula for the difference of squares is \[ (a - b)(a + b) = a^2 - b^2 \].

Apply the special product formula

In this exercise, \( a = x \) and \( b = 1 \). Using the difference of squares formula: \[ (x-1)(x+1) = x^2 - 1^2 \].

Simplify the expression

Simplify \( x^2 - 1^2 \) to get \( x^2 - 1 \).

Key concepts

Difference of Squares

The 'difference of squares' is a specific type of polynomial multiplication that results in a simplified expression. The general formula for the difference of squares is: (a - b)(a + b) = a^2 - b^2 It shows that the product of two binomials, where one binomial has subtraction and the other addition, equals the difference between the square of the first term and the square of the second term. In our given exercise, we see this formula at work with: (x - 1)(x + 1) Here, we identify 'a' as 'x' and 'b' as 1, resulting in: (x - 1)(x + 1) = x^2 - 1^2 = x^2 - 1. Understanding this formula helps to quickly and accurately solve such problems.

Special Product Formulas

Special product formulas are shortcuts that simplify the multiplication of polynomials. Instead of expanding every term, these formulas provide a quick way to find the product. Three main special product formulas are often used:

• Square of a Binomial: (a + b)^2 = a^2 + 2ab + b^2.
• Square of a Binomial (subtraction): (a - b)^2 = a^2 - 2ab + b^2.
• Difference of Squares: (a - b)(a + b) = a^2 - b^2.

In the given problem, we used the 'difference of squares' formula: (x - 1)(x + 1) By recognizing and applying these formulas correctly, we can simplify complex polynomial multiplications efficiently.

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in the other polynomial. For example, if we have (a + b)(c + d), we multiply each term: a*c + a*d + b*c + b*d. However, special product formulas can simplify this process. When we applied the 'difference of squares' formula, instead of multiplying each term individually, we used: (x - 1)(x + 1) and directly arrived at x^2 - 1^2 = x^2 - 1. This saved us time while ensuring accuracy.

Standard Form

The standard form of a polynomial is a way of writing the polynomial with terms in descending order of their degrees. For example, a standard form of a polynomial can be: ax^2 + bx + c. In our given problem, after applying the difference of squares formula, we obtained x^2 - 1. This is already in standard form because it is written as: x^2 + 0x - 1. Expressing polynomials in standard form helps in organizing and simplifying the expressions clearly.