Problem 31 - Elementary Algebra

Question

For the following problems, find the products. $$ (5 a-3 b)^{2} $$

Step-by-Step Solution

Answer

Question: Find the product of the expression \((5a - 3b)^2\). Answer: The product of the given expression is \((5a - 3b)^2 = 25a^2 - 30ab + 9b^2\).

Step 1: Write the binomial squared as a product

Since we are given the expression \((5a - 3b)^2\), we can rewrite it as \((5a - 3b)(5a - 3b)\) for better understanding of what's happening in this problem.

Step 2: Apply the FOIL method

Now we can apply the FOIL method (First, Outer, Inner, Last) to find the product of our expression. Begin by multiplying each term of the first binomial by each term in the second binomial.

Step 3: Perform the multiplications

Now we have to perform the multiplications. - First: \((5a)(5a)\) - Outer: \((5a)(-3b)\) - Inner: \((-3b)(5a)\) - Last: \((-3b)(-3b)\)

Step 4: Simplify each product

Next, simplify each of the products obtained in step 3. - First: \((5a)(5a) = 25a^2\) - Outer: \((5a)(-3b) = -15ab\) - Inner: \((-3b)(5a) = -15ab\) - Last: \((-3b)(-3b) = 9b^2\)

Step 5: Combine like terms and write the final expression

Now combine the like terms obtained in step 4: \(25a^2 -15ab -15ab +9b^2\) Combine the middle terms: \(25a^2 - 30ab + 9b^2\) So, the product of the given expression is: $$(5a - 3b)^2 = 25a^2 - 30ab + 9b^2$$

Key Concepts

Binomial expansion

Binomial expansion is a critical concept in algebra that helps in simplifying expressions raised to a power. In the context of our exercise, we are expanding \((5a - 3b)^2\). The term "binomial" refers to an expression composed of two terms, like \(5a - 3b\). Expanding such expressions is affectionately known as the "binomial expansion".
The binomial expansion involves rewriting the squared binomial as the product \((5a - 3b)\times(5a - 3b)\). This approach simplifies calculations, allowing you to apply systematic multiplication rather than directly squaring individual terms, which can often be error-prone.
When you expand an expression like \((x + y)^n\), you end up with a more extended polynomial. The polynomial is created by applying distributive properties and combining like terms. Mastering binomial expansion allows for effective problem-solving in algebra, particularly in simplifying polynomials.

FOIL method

The FOIL method is a mnemonic that simplifies the process of multiplying two binomials. FOIL stands for First, Outer, Inner, Last, reflecting the order of operations in the multiplication process. This method is particularly helpful because it breaks down multiplication into manageable steps. Let's break down the FOIL method using our example, \((5a - 3b)(5a - 3b)\):

• First: Multiply the first terms in each binomial: \((5a)\times(5a) = 25a^2\)
• Outer: Multiply the outer terms: \((5a)\times(-3b) = -15ab\)
• Inner: Multiply the inner terms: \((-3b)\times(5a) = -15ab\)
• Last: Multiply the last terms in each binomial: \((-3b)\times(-3b) = 9b^2\)

By breaking down the product of the binomials into these four steps, FOIL ensures each part of the expression is accounted for. After completing these multiplications, the next step is to combine like terms, resulting in the simplified expression: \(25a^2 - 30ab + 9b^2\). The FOIL method is indispensable for students learning polynomial multiplication, as it simplifies complex calculations into a series of simple tasks.

Polynomial multiplication

Polynomial multiplication involves general strategies for multiplying expressions with multiple terms, expanding beyond simple binomial products. The concept of multiplying polynomials is central to algebra, as it is used for expanding expressions, solving equations, and in calculus for calculus operations like differentiation and integration.
In our exercise, the multiplication process for the binomial \((5a - 3b)\) involves calculating the product with itself. The process, demonstrated through the use of the FOIL method, consists of distributing each term of the first polynomial by every term of the second polynomial.
After applying the FOIL method, we end with several terms: \(25a^2\), two terms of \(-15ab\), and \(9b^2\). Polynomial multiplication includes the important step of combining like terms—in this case, combining the like terms of \(-15ab\) to get \(-30ab\). The final outcome of our multiplication task is: \[25a^2 - 30ab + 9b^2\]Being skilled at polynomial multiplication allows students to tackle more advanced algebraic problems with confidence, knowing they can manage expressions of varying complexity efficiently.