Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the polynomials. \((4 t-5 u)^{2}\)

Step-by-Step Solution

Answer

\((4t - 5u)^2 = 16t^2 - 40tu + 25u^2\).

Step 1: Recognize the Pattern

Identify the expression as a binomial squared: \((4t - 5u)^2\). The square of a binomial follows the pattern \((a - b)^2 = a^2 - 2ab + b^2\).

Step 2: Identify Components

In the binomial \((4t - 5u)^2\), identify \(a = 4t\) and \(b = 5u\).

Step 3: Square the First Term

Calculate \(a^2\) which is \((4t)^2 = 16t^2\).

Step 4: Calculate the Double Product

Calculate \(-2ab\): \(-2 \times 4t \times 5u = -40tu\).

Step 5: Square the Second Term

Calculate \(b^2\) which is \((5u)^2 = 25u^2\).

Step 6: Combine the Results

Combine all the pieces to get the expanded polynomial: \(16t^2 - 40tu + 25u^2\).

Key Concepts

Binomial Expansion

When you come across a binomial expression raised to a power, such as \[(4t - 5u)^2,\] binomial expansion comes to the rescue. This technique helps simplify expressions by expanding them into a polynomial.

Binomials are algebraic expressions that have two terms. The expression above is a binomial, and when squared, it fits into a commonly used pattern:

• \((a - b)^2 = a^2 - 2ab + b^2\)

This formula is the key to expanding any squared binomial expression.

To break it down:

• First, square the first term \(a^2\).
• Next, calculate \(-2ab\), which involves multiplying the two terms and including a negative sign.
• Finally, square the second term \(b^2\).

Put it all together, and you have your expanded polynomial. This method not only saves time but also reduces the chances of making errors compared to manually multiplying the binomials.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They serve as the foundation for many mathematical concepts. When dealing with expressions like \((4t - 5u)\), it's crucial to understand how to manipulate and simplify them.

In algebra, expressions can:

• Represent real-world situations or mathematical relationships.
• Be added, subtracted, multiplied, or divided like numbers.

With algebraic techniques, you can transform an expression into a form that's easier to handle. For example, multiplying \((4t - 5u)^2\) not only involves applying binomial expansion but understanding the interactions of the variables \(t\) and \(u\) in depth.

Algebraic expressions bridge the gap between abstract math and practical applications, allowing you to model and solve problems in diverse areas.

Quadratic Polynomials

Quadratic polynomials are a type of polynomial characterized by their highest degree of two. This means the variable is squared, and such a polynomial typically takes the form:\[ax^2 + bx + c\]where "\(a\), \(b\)," and "\(c\)" are constants. In our example, the result of expanding \((4t - 5u)^2\) yields a quadratic polynomial:\[16t^2 - 40tu + 25u^2\]This expression contains three terms, each representing a part of the binomial expansion.

• \(16t^2\) is the square of the first term.
• \(-40tu\) is the double product of the two terms.
• \(25u^2\) is the square of the second term.

Quadratic polynomials can be solved or factored, and they show up often in equations involving area, projectile motion, and numerous other scenarios in physics and engineering. Understanding how to expand and simplify them is fundamental in algebra and beyond.