Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \left(3 x+y^{3}\right)\left(3 x-y^{3}\right) $$

Step-by-Step Solution

Answer

The expression is simplified to \(9x^2 - y^6\).

Step 1: Identify the Expression Type

The expression \((3x + y^3)(3x - y^3)\) is in the form \((a + b)(a - b)\), which is a difference of squares.

Step 2: Apply Difference of Squares Formula

The formula for the difference of squares is \((a + b)(a - b) = a^2 - b^2\). In our case, assign \(a = 3x\) and \(b = y^3\).

Step 3: Square Each Term

Calculate \(a^2 = (3x)^2 = 9x^2\) and \(b^2 = (y^3)^2 = y^6\).

Step 4: Subtract the Squares

Substitute the squared terms into the difference of squares formula: \(a^2 - b^2 = 9x^2 - y^6\).

Key Concepts

Difference of Squares

The difference of squares is a fundamental concept in polynomial mathematics. It involves transforming an expression written as a product of two binomials, such as

• \[(a + b)(a - b)\]

into a simpler form.
This pattern indicates that the product will result in the subtraction of squares. The formula for this is:

• \[(a + b)(a - b) = a^2 - b^2\]

The terms to the left,

• \((a + b)\)
• \((a - b)\)

are called conjugates, and the subtraction results in the difference of their respective squares.
In the case of our original expression

• \((3x + y^3)(3x - y^3)\)

we identify:

• \(a = 3x\)
• \(b = y^3\)

By squaring each, we apply the pattern to get

• \(9x^2 - y^6\)

as the simplified form.

Polynomial Multiplication

Polynomial multiplication is a crucial operation when dealing with algebraic expressions. The key is to ensure each term in one polynomial is multiplied by every term in the other. This operation is often facilitated by identifying patterns or using formal methods like the distributive property.
Considering our example

• \((3x + y^3)(3x - y^3)\)

this is more efficiently handled using the difference of squares method rather than mere expansion. However, understanding multiplication is essential:- Multiply each term in the first binomial by each term in the second binomial.- Combine like terms to simplify.
When expanded directly, this would involve calculations like

• \( (3x)(3x) = 9x^2 \)
• \( (3x)(-y^3) = -3xy^3 \)
• \( (y^3)(3x) = 3xy^3 \)
• \( (y^3)(-y^3) = -y^6 \)

These terms would be combined to show that intermediate terms cancel, thus simplifying to

• \(9x^2 - y^6\)

without unnecessary complexity.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators. They form the basis of most algebra operations, as seen in polynomial mathematics. An expression like

• \((3x + y^3)(3x - y^3)\)

is crafted using various terms. Each term in an expression can be labeled as either:

• Constant: A fixed number.
• Variable: Symbol representing an unknown number, like \(x\) or \(y\).
• Coefficient: A number multiplying a variable, such as 3 in \(3x\).

In polynomial operations, recognizing components of expressions is vital. This understanding allows you to:- Efficiently apply rules like the difference of squares.- Navigate more complex algebra with ease.
Ultimately, comprehension of these structures equips you to handle more intricate mathematical problems. When dealing with algebraic expressions, always pay attention to the arrangement and relationship of terms to simplify effectively without error.