Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (x+3 y)(x-3 y) $$

Short answer

The product is \( x^2 - 9y^2 \).

Step-by-step solution

Recognize the Special Product Formula

The expression \( (x+3y)(x-3y) \) matches the form of the difference of squares formula, which is \( (a+b)(a-b) = a^2 - b^2 \). Here, \( a = x \) and \( b = 3y \).

Apply the Difference of Squares Formula

Using the difference of squares formula: \( (x+3y)(x-3y) = x^2 - (3y)^2 \).

Simplify the Expression

Simplify the squared term: \( x^2 - (3y)^2 = x^2 - 9y^2 \).

Key concepts

special product formulas

Special product formulas are powerful tools in algebra that make expanding certain polynomial products much easier. One such formula is the difference of squares formula. This helps simplify expressions like \( (a+b)(a-b) \) into \( a^2 - b^2 \). This formula is particularly useful because it avoids the need for lengthy FOIL (First, Outer, Inner, Last) multiplication. Using this shortcut can save time and effort, especially with more complex polynomials.

polynomial multiplication

Polynomial multiplication is a key technique in algebra, used to find the product of two or more polynomials. Typically, we can multiply polynomials by using the distributive property. However, special product formulas, like the difference of squares, streamline the process. For instance, in the expression \( (x+3y)(x-3y) \), using the difference of squares formula \( (a+b)(a-b) = a^2 - b^2 \) simplifies it to \( x^2 - (3y)^2 \), which further simplifies to \( x^2 - 9y^2 \). By recognizing these patterns, polynomial multiplication becomes quicker and more efficient.

standard form polynomial

A polynomial in standard form is expressed in terms of descending powers of its variable. For example, \( x^2 - 9y^2 \) is in standard form because the terms are ordered from highest to lowest degree. This form is useful for identifying the structure and degree of the polynomial, which helps in solving and graphing equations. Transforming your expression into standard form eliminates confusion and ensures consistency in equations and further calculations.