Question

Multiply the polynomials using the special product formulas. Express your answer as a single polynomial in standard form. $$ (5 x-3)(5 x+3) $$

Short answer

25x^2 - 9

Step-by-step solution

Identify the special product formula

Notice that the given expression \( (5x - 3)(5x + 3) \) fits the form of the difference of squares. Recall the special product formula for the difference of squares: \[ (a - b)(a + b) = a^2 - b^2 \]

Apply the special product formula

In this case, \(a = 5x \) and \(b = 3 \). Substitute these values into the formula: \[ (5x - 3)(5x + 3) = (5x)^2 - 3^2 \]

Simplify the expression

Calculate the squares: \[ (5x)^2 = 25x^2 \] and \[ 3^2 = 9 \]. So the expression becomes \[ 25x^2 - 9 \]

Write the final polynomial in standard form

The polynomial \[ 25x^2 - 9 \] is already in standard form because it is written as \[ax^2 + bx + c \] with terms in descending order of the powers of \(x \).

Key concepts

Special Product Formulas

Special product formulas are shortcuts that simplify the multiplication of polynomials. They help recognize patterns, saving you from lengthier multiplication steps.

One of the most commonly used special product formulas is the difference of squares. It states that \((a - b)(a + b) = a^2 - b^2 \).

This means that if you have two binomials where the first terms are the same and the second terms are opposites, you can directly use this formula to get a simplified form.

Tips for using special product formulas:

• Identify the pattern--look for matching and opposite terms.
• Substitute into the formula directly.
• Simplify the result by performing the indicated operations.

Understanding these patterns can make polynomial multiplication quicker and more accurate.

Difference of Squares

The difference of squares is one of the special product formulas. It is represented as \((a - b)(a + b) = a^2 - b^2 \).

Here, you'll multiply two binomials that only differ in the sign of their second term.

Let's break it down using the exercise \((5x - 3)(5x + 3) \):

• In this case, \( a = 5x \) and \( b = 3 \)
• Applying the difference of squares formula, we get \( (5x)^2 - 3^2 \)
• Simplifying gives us \( 25x^2 - 9 \).

The key advantage here is skipping the direct term-by-term multiplication and leveraging the formula for an immediate result.

Standard Form of Polynomials

A polynomial is in standard form when its terms are ordered in descending powers of the variable.

For our example, \(25x^2 - 9 \), is already in standard form.

Here's why:

• The highest power term \(25x^2 \) comes first.
• The constant term \(-9 \) is placed at the end.

The standard form ensures consistency and makes comparison and further operations with polynomials easier. Remember to always align the terms from the highest to the lowest degree to maintain the standard form.