Question

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

Short answer

9x^2 - 16y^2

Step-by-step solution

Identify the Special Product Formula

The given polynomials \(3x + 4y\) and \(3x - 4y\) can be multiplied using the difference of squares formula. This formula states that \((a + b)(a - b) = a^2 - b^2\).

Define Variables

Identify \(a\) and \(b\) from the polynomials. Here, \(a = 3x\) and \(b = 4y\).

Apply the Difference of Squares Formula

Substitute \(a = 3x\) and \(b = 4y\) into the formula \((a + b)(a - b) = a^2 - b^2\). This gives \(a^2 - b^2 = (3x)^2 - (4y)^2\).

Simplify the Expression

Calculate the squares of \(a\) and \(b\). \( (3x)^2 = 9x^2\) and \( (4y)^2=16y^2 \).

Final Expression

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

Key concepts

Difference of Squares

In algebra, the difference of squares is a special product formula that simplifies the multiplication of certain binomials. The formula is \(a^2 - b^2 = (a + b)(a - b)\).
It's handy because rather than multiplying binomials directly, you recognize a pattern and use this formula to reach the answer quickly.
The expression \(3x + 4y\) and \(3x - 4y\) fits this pattern. Here, \(a\) is \(3x\)and \(b\) is \(4y\). By knowing the difference of squares formula, we can simplify the problem easily.

Polynomial Multiplication

Polynomial multiplication can be simplified using special product formulas like the difference of squares.
When multiplying \( (3x+4y)(3x-4y) \), directly applying the difference of squares formula avoids the longer method of distributive property.
Here, recognizing the pattern \(a^2 - b^2\) allows us to quickly move to squaring the terms: \(9x^2\) and \(16y^2\).
Thus, using special products like this makes solving polynomial multiplications easier and less time-consuming.

Standard Form of Polynomial

The standard form of a polynomial means writing it in a way where terms are ordered by decreasing degree.
After applying the difference of squares to \( (3x+4y)(3x-4y) \), we reach \( 9x^2 - 16y^2 \).
Here, \( 9x^2 \) and \( 16y^2 \) are already squared terms, and writing them in descending order of degrees (exponents) is necessary for the standard form.
This format helps in recognizing and further manipulating polynomials in algebra.

Simplifying Expressions

Simplifying expressions means making them as concise as possible.
Using the special product formula, we quickly simplify \( (3x+4y)(3x-4y)\) to \( 9x^2 - 16y^2 \).
This avoids the more cumbersome step-by-step multiplication and combination of like terms.
Simplifying makes the final expression easier to handle for further calculations or uses in different equations.
It helps in reducing potential errors and keeps expressions manageable in complex problem solving.