Problem 33 - Elementary Algebra

Question

For the following problems, find the products. $$ (2 h-8 k)^{2} $$

Step-by-Step Solution

Answer

Answer: The product of the expression $(2h-8k)^2$ is $4h^2 - 32hk + 64k^2$.

Step 1: Write down the given expression

Write the expression: $$ (2h-8k)^2 $$

Step 2: Rewrite the expression

Rewrite the expression as a product of the binomial with itself: $$ (2h-8k)(2h-8k) $$

Step 3: Apply the distributive property or FOIL method

Multiply the first terms, the outer terms, the inner terms, and the last terms: First: \((2h)(2h)\) Outer: \((2h)(-8k)\) Inner: \((-8k)(2h)\) Last: \((-8k)(-8k)\)

Step 4: Multiply each term

Calculate the product for each term: First: \((2h)(2h) = 4h^2\) Outer: \((2h)(-8k) = -16hk\) Inner: \((-8k)(2h) = -16hk\) Last: \((-8k)(-8k) = 64k^2\)

Step 5: Combine the terms

Combine all the products obtained in step 4: $$ 4h^2 - 16hk - 16hk + 64k^2 $$

Step 6: Simplify the expression

Simplify by combining like terms: $$ 4h^2 - 32hk + 64k^2 $$ The simplified product of the given expression is: $$ (2h-8k)^2 = 4h^2 - 32hk + 64k^2 $$

Key Concepts

Polynomials

Polynomials are expressions that consist of variables and coefficients, connected by addition, subtraction, and multiplication. Each individual term in a polynomial features a product of a constant and a variable raised to a whole number power. In the expression \((2h-8k)^2\), we are dealing with a binomial, which is a type of polynomial consisting of exactly two terms: \(2h\) and \(-8k\). The act of squaring the binomial in this problem entails multiplying it by itself, which transforms it into a quadratic polynomial.When a binomial is squared, the result is typically a trinomial if simplified, containing three main terms – as seen in the result \(4h^2 - 32hk + 64k^2\). Understanding how to manipulate these terms is crucial for mastering polynomial expressions and solving problems in algebra.

Distributive Property

The distributive property is a fundamental principle used to simplify expressions and is especially handy in expanding and multiplying polynomials. It states that for any numbers or expressions \(a\), \(b\), and \(c\):\[ a(b + c) = ab + ac \]In the context of this problem, after rewriting the square of the binomial as \((2h-8k)(2h-8k)\), we use the distributive property to systematically multiply each term in the first binomial by each term in the second binomial.This yields four distinct products that need to be calculated and then combined:

• First: \((2h)(2h)\) resulting in \(4h^2\)
• Outer: \((2h)(-8k)\) giving \(-16hk\)
• Inner: \((-8k)(2h)\) also \(-16hk\)
• Last: \((-8k)(-8k)\) resulting in \(64k^2\)

Combining these products according to the distributive property helps in forming the final simplified expression: \(4h^2 - 32hk + 64k^2\).

FOIL Method

The FOIL method is a shortcut useful for remembering the steps to expand the product of two binomials. It stands for First, Outer, Inner, Last, referencing the terms to be multiplied:- **First:** Multiply the first terms in each binomial: \((2h)(2h) = 4h^2\)- **Outer:** Multiply the outer terms: \((2h)(-8k) = -16hk\)- **Inner:** Multiply the inner terms: \((-8k)(2h) = -16hk\)- **Last:** Multiply the last terms: \((-8k)(-8k) = 64k^2\)Using the FOIL method ensures that you account for all parts of the binomials, resulting in an expanded expression. After calculating these terms, they must be combined and simplified. This simplification involves techniques like combining like terms, here combining \(-16hk\) and \(-16hk\) to achieve \(-32hk\). Thus, the FOIL method serves as a strategic approach to simplifying the multiplication of binomials, as seen in our final polynomial: \(4h^2 - 32hk + 64k^2\).