Problem 65 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$16 x^{2}-40 x y+25 y^{2}$$

Step-by-Step Solution

Answer

The factorization of the given perfect square trinomial \(16x^{2}-40xy+25y^{2}\) is \((4x-5y)^{2}\)

Step 1: Inspect the terms

Observe the given polynomial \(16x^{2}-40xy+25y^{2}\). In comparison to the general perfect square trinomial \(a^{2}-2ab+b^{2}\), it can be seen that \(a^{2} = 16x^{2}\), \(-2ab = -40xy\) and \(b^{2} = 25y^{2}\)

Step 2: Find the square roots

Find the square root of \(a^{2}\) and \(b^{2}\) to get 'a' and 'b' respectively. For \(a^{2} = 16x^{2}\), 'a' is 4x. For \(b^{2} = 25y^{2}\), 'b' is 5y.

Step 3: Check validity

To ensure the terms are correct, compute \(2*a*b\), which should equal to the second term of the trinomial (-40xy).

Step 4: Write down the final factorization

If the terms satisfy the rules of perfect square trinomials, then we can write down the factorization as \((a-b)^{2}\). In our case, this would be \((4x-5y)^{2}\).

Key Concepts

Polynomial Factorization

Factoring polynomials is an essential technique in algebra that simplifies expressions by expressing them as a product of their factors. This is especially useful for solving equations and analyzing their roots. For a polynomial to be factored, it usually means rewriting it by identifying and separating common factors. There are several methods to factor polynomials, including:

• Extracting common factors.
• Grouping terms.
• Factoring quadratics into binomials.
• Recognizing special products like perfect square trinomials and difference of squares.

In the given exercise, recognizing the polynomial as a perfect square trinomial allows it to be easily factored into a squared binomial. This characteristic is vital for efficient problem-solving, showing the elegance of algebraic manipulation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the basic building blocks of algebra, representing real-world quantities and relationships.In the context of our problem, the expression \(16x^{2} - 40xy + 25y^{2}\) is an example of a polynomial, where:

• Coefficients are the numerical parts, like 16, -40, and 25.
• Variables are the alphabetic representations like \(x\) and \(y\).
• Terms are the separate additive components of the expression, like \(16x^{2}\), \(-40xy\), and \(25y^{2}\).

Each part plays a role in simplifying or restructuring the expressions, which is necessary for factorization or solving equations. Understanding the nuances of algebraic expressions is crucial for transforming and utilizing them in various mathematical contexts.

Square Roots

Square roots are one of the fundamental operations in algebra, necessary for solving many equations, and are particularly significant in the factorization of perfect square trinomials.A square root of a number \(n\) is a value that, when multiplied by itself, gives \(n\). In the context of our exercise, finding the square roots of \(16x^{2}\) and \(25y^{2}\) was key:

• The square root of \(16x^{2}\) is \(4x\), since \((4x) \times (4x) = 16x^{2}\).
• The square root of \(25y^{2}\) is \(5y\), since \((5y) \times (5y) = 25y^{2}\).

Determining these square roots helps identify the values needed to form the binomial expression that results from factorizing a perfect square trinomial.

Trinomial Structure

The structure of a trinomial is composed of three terms, usually expressed in the form \(ax^2 + bx + c\). Understanding this structure is crucial to recognizing perfect square trinomials and simplifying them.For the trinomial given in the exercise, identify it as a perfect square trinomial by matching it to the form \(a^2 - 2ab + b^2\). In this case:

• \(a^2 = 16x^{2}\)
• \(-2ab = -40xy\)
• \(b^2 = 25y^{2}\)

When these terms fit the required format, it allows the trinomial to be expressed as \((a-b)^2\), simplifying it significantly. Recognizing this special structure is a key insight in algebra, saving both time and effort in problem-solving.