Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (3 x+4)(3 x-4) $$

Short answer

9x^2 - 16

Step-by-step solution

- Identify the Special Product Formula

Recognize that the expression \( (3x + 4)(3x - 4) \) fits the form of the difference of squares formula: \( (a + b)(a - b) = a^2 - b^2 \). Here, \( a = 3x \) and \( b = 4 \).

- Apply the Difference of Squares Formula

Using the formula: \( (3x + 4)(3x - 4) = (3x)^2 - 4^2 \).

- Simplify Each Term

Calculate \( (3x)^2 \) and \( 4^2 \): \( (3x)^2 = 9x^2 \) and \( 4^2 = 16 \).

- Form the Polynomial

Combine the results to express the polynomial in standard form: \( 9x^2 - 16 \).

Key concepts

Special Product Formulas

Special product formulas are shortcuts for multiplying certain types of polynomials. These formulas save time and reduce errors. The most common special product formulas are the difference of squares, perfect square trinomials, and sum and difference of cubes.

In this exercise, we used the difference of squares formula. Other situations might call for different formulas. For example:

• Perfect square trinomial: \((a + b)^2 = a^2 + 2ab + b^2\)
• Sum and difference of cubes: \((a^3 + b^3) = (a + b)(a^2 - ab + b^2)\)

Knowing these formulas helps simplify polynomial multiplication.

Difference of Squares

The difference of squares formula is used to simplify the multiplication of two binomials of the form \((a + b)(a - b)\). It states that \((a + b)(a - b) = a^2 - b^2\).

Here's an example from the exercise:

• The original expression was \((3x + 4)(3x - 4)\)
• Identify \((a = 3x)\) and \((b = 4)\)
• Apply the formula to get \((a^2 - b^2)\)
• This results in \((\( (3x)^2 - 4^2 \))\)
• Simplify to \((9x^2 - 16)\)

Using this form saves time and makes polynomial multiplication easier.

Standard Form Polynomial

A polynomial is in standard form when its terms are written in descending order of their degrees. The degrees indicate the power of the variable in each term.

For example, in the polynomial \((9x^2 - 16)\):

• \(9x^2\) is the term with the highest degree (2).
• \(-16\) is the constant term with degree 0.

Writing polynomials in standard form makes it easier to read and understand. It's also helpful for comparing polynomials.

When you multiply polynomials using special product formulas, the final expression should be in standard form. This helps ensure clarity and accuracy in your work.