Problem 6 - Precalculus: Functions and Graphs

Question

Express as a polynomial. $$(5 x+4 y)(5 x-4 y)$$

Step-by-Step Solution

Answer

The polynomial expression is \(25x^2 - 16y^2\).

Step 1: Identify the Expression Type

The expression \((5x + 4y)(5x - 4y)\) is a difference of squares. This is a standard form where \((a + b)(a - b) = a^2 - b^2\).

Step 2: Assign Values to a and b

Identify \(a\) and \(b\) in the expression \((a + b)(a - b)\). Here, \(a = 5x\) and \(b = 4y\).

Step 3: Apply the Difference of Squares Formula

According to the difference of squares formula, \((a + b)(a - b) = a^2 - b^2\). Substitute \(a = 5x\) and \(b = 4y\) into the formula to get \((5x)^2 - (4y)^2\).

Step 4: Calculate \(a^2\) and \(b^2\)

Calculate \((5x)^2 = 25x^2\) and \((4y)^2 = 16y^2\).

Step 5: Form the Polynomial

Substitute the values found in Step 4 into \(a^2 - b^2\), giving the polynomial \(25x^2 - 16y^2\).

Key Concepts

Difference of Squares

The term "difference of squares" refers to a specific type of polynomial expression that can be factored using a simple pattern. When you have two terms that are both perfect squares, separated by a subtraction sign, you can express this as a difference of squares. The general rule is that \((a + b)(a - b) = a^2 - b^2\). This pattern is extremely useful because it allows you to simplify expressions quickly.
For example, in our exercise, the expression \((5x + 4y)(5x - 4y)\) is identified as a difference of squares. Here, both \(5x\) and \(4y\) are treated as the squares. The subtraction sign between them indicates the "difference," setting up for this straightforward factorization.
Recognizing patterns and knowing the formulas can save time and make algebra much easier.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, or simpler polynomials. This process is crucial in solving equations, simplifying expressions, and understanding the structure of polynomials.
To factor a polynomial like a difference of squares, find two binomials that, when multiplied, give you the original expression. Our original problem \((5x + 4y)(5x - 4y)\) is already in a factored form.
Factoring helps in solving equations and analyzing polynomials because it breaks down complex expressions into more manageable parts.

Product of Binomials

A product of binomials involves multiplying two binomial expressions, which are algebraic expressions with two terms. Binomials can often be multiplied using the FOIL method, which stands for First, Outer, Inner, Last — referring to the terms in each binomial that you multiply together.
However, when we deal with the special case of the difference of squares, you simply apply the formula \((a + b)(a - b) = a^2 - b^2\). This shortcut saves time and avoids the need to perform each multiplication separately. In practice, notice that the terms \(5x + 4y\) and \(5x - 4y\) multiply directly to give \(25x^2 - 16y^2\) without any additional steps.

Polynomial Expansion

Polynomial expansion refers to breaking down an expression that is raised to a power or multiplied into a sum of terms. It includes aspects of distributing terms and combining like terms.
In our exercise, expanding happens through the application of the difference of squares formula. Here, the multiplication of \((5x+4y)(5x-4y)\) expands into \(25x^2 - 16y^2\), a simpler expression.
Expansion helps to see the full form of a polynomial expression, making it easier to analyze and solve. By understanding polynomial expansion, you get insights into the structure and behavior of polynomials, which is vital for solving advanced mathematics problems.