Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (x-7)(x+7) $$

Short answer

x^2 - 49

Step-by-step solution

Identify the special product formula

The given expression \( (x-7)(x+7) \) can be identified as the difference of squares formula, which is \[ (a-b)(a+b) = a^2 - b^2. \]

Apply the difference of squares formula

In the expression \( (x-7)(x+7) \), recognize that \( a = x \) and \( b = 7 \). Substitute these into the formula to get \[ x^2 - 7^2. \]

Simplify the expression

Calculate \( 7^2 \), which is 49. Therefore, the expression becomes \[ x^2 - 49. \]

Write the final answer in standard form

Since \[ x^2 - 49 \] is already in standard form, the final answer is \[ x^2 - 49. \]

Key concepts

Difference of Squares

When we talk about polynomials, one interesting concept is the difference of squares. This occurs when you multiply two binomials that are similar except for the sign between them. For example, the binomials in the exercise were \(x - 7\) and \(x + 7\). The difference of squares formula is \[ (a-b)(a+b) = a^2 - b^2. \]

Here, \(a = x\) and \(b = 7\). When you apply this formula to our binomials, we substitute in for \(x\) and \(7\), leading us to \[x^2 - 7^2. \] Thus, multiplying \(x - 7\) and \(x + 7\) results in the difference of squares.

Simplifying further, \(7^2 = 49\). So the expression becomes \[ x^2 - 49. \]

Special Product Formulas

Special product formulas are shortcuts that make polynomial multiplication easier. There are several of these formulas:

• \textbf{Square of a Binomial:} \[ (a + b)^2 = a^2 + 2ab + b^2 \] and \[ (a - b)^2 = a^2 - 2ab + b^2. \]
• \textbf{Difference of Squares:} The one we used in the exercise, \[ (a - b)(a + b) = a^2 - b^2. \]
• \textbf{Product of a Sum and a Difference:} \[ (a+b)(a-b) = a^2 - b^2. \]

Using these formulas can save time and reduce errors when multiplying polynomials. They allow you to directly arrive at the simplified form without having to multiply each term individually.

Standard Form

Standard form is simply a way to write polynomials where the terms are ordered from the highest degree to the lowest degree. For example, a polynomial in standard form looks like: \[ ax^2 + bx + c. \]

In the provided exercise, our final answer, \[ x^2 - 49, \] is already in standard form. The degree of the polynomial is determined by the highest power of its variable (in this case, \(x^2\)). Each subsequent term's degree should be in descending order. Since there are no x terms, \[ x^2 - 49 \] is correct as is.

Writing polynomials in standard form ensures consistency and makes it easier to perform operations like addition, subtraction, and further multiplication involving other polynomials.