Problem 66 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$9 x^{2}+48 x y+64 y^{2}$$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(9x^{2} + 48xy + 64y^{2}\) is \((3x + 8y)^{2}\).

Step 1: Identify the Squares

First, identify the square of the first and last term. So, \(9x^{2}\) can be written as \((3x)^{2}\), and \(64y^{2}\) can be written as \((8y)^{2}\).

Step 2: Checking the middle term

Check if the middle term is twice the product of the square root of the first term and the square root of the last term. In this case, \(48xy\) is equal to \(2 * 3x * 8y\).

Step 3: Factorization

The trinomial can be factorized as \((3x + 8y)^{2}\). This means that \(9x^{2} + 48xy + 64y^{2} = (3x + 8y)^{2}\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler terms called factors. This process allows us to express the polynomial as a product of two or more polynomials. Having a solid understanding of this helps solve equations more easily.

When working with perfect square trinomials, identifying them is crucial. A trinomial of the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\) can be expressed as \((a+b)^2\) or \((a-b)^2\) respectively. Recognizing these forms allows for quick factorization, turning complex expressions into manageable factors.

Here’s a quick tip for factoring:

• Identify the first and last terms as perfect squares.
• Ensure the middle term is twice the product of the square roots of the first and last terms.

Mastering this technique can significantly simplify solving polynomial equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Understanding these expressions is foundational in algebra as they form the building blocks of equations and inequalities.

In our exercise, the algebraic expression is a trinomial: \(9x^2 + 48xy + 64y^2\). Each term in this expression is separated by an addition or subtraction sign. The expression includes coefficients and variables:

• 9 is the coefficient of \(x^2\)
• 48 is the coefficient of \(xy\)
• 64 is the coefficient of \(y^2\)

Algebraic expressions can be simplified or factored, allowing us to solve equations or find their values for given variables. Understanding how to manipulate and factor these expressions is key to mastering algebra.

Polynomial Factorization

Polynomial factorization is the process of decomposing a polynomial into a product of other polynomials that, when multiplied together, give the original polynomial. This is especially useful in simplifying expressions and solving polynomial equations.

In our given exercise, we're dealing with a specific type of polynomial, the perfect square trinomial. The steps to factor it include:

• Identifying the squares in the first and last terms (e.g., \((3x)^2\) and \((8y)^2\))
• Ensuring the middle term matches the requirement of being twice the product of the roots of the first and last terms (e.g., \(2 \times 3x \times 8y = 48xy\))
• Finally rewriting the expression as the square of a binomial: \((3x + 8y)^2\)

By following these steps, polynomial factorization becomes a manageable process, turning complex polynomials into simpler expressions that reveal more about their nature and potential solutions.