Problem 17 - Elementary Algebra

Question

For the following problems, find the products. $$ (a-7)^{2} $$

Step-by-Step Solution

Answer

Answer: The product of (a-7)^2 is a^2 - 14a + 49.

Step 1: Understand the given problem

We are given the expression (a-7)^2, which means we need to multiply (a-7) by itself.

Step 2: Apply the multiplication method

Multiply (a-7) by itself which can be written as: (a-7)(a-7)

Step 3: Distribute the terms

In order to multiply these terms, we need to distribute each term of the first expression to each term of the second expression, following the FOIL method: First terms: a * a = a^2 Outer terms: a * (-7) = -7a Inner terms: (-7) * a = -7a Last terms: (-7) * (-7) = 49

Step 4: Combine like terms

Combine the like terms obtained from the distribution process: a^2 - 7a - 7a + 49

Step 5: Simplify the expression

Combine the like terms: a^2 - 14a + 49 So, the product of (a-7)^2 is: $$ (a-7)^2 = a^2 - 14a + 49 $$

Key Concepts

Multiplication of Polynomials

Multiplying polynomials involves combining two algebraic expressions to produce a single polynomial. This can be as simple as multiplying monomials (single terms) or more complex with binomials or trinomials. In our exercise, we are dealing with a binomial, \( (a-7)^2 \), which requires multiplying the expression by itself.

To achieve this, every term in the first binomial is multiplied by every term in the second binomial. The result is a set of terms that we must combine by addition or subtraction. This process is crucial when moving forward in algebra, as the same principles apply to more complicated polynomials.

FOIL Method

The FOIL method is a special technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, which represents the order in which you multiply the terms of the binomials.

- **First**: Multiply the first terms in each binomial.- **Outer**: Multiply the outer terms in the product.- **Inner**: Multiply the inner terms.- **Last**: Multiply the last terms in each binomial.

The method streamlines the multiplication process and ensures every part of the binomial is addressed. This step-by-step process helps students understand how each term contributes to the final expanded expression. For our problem, applying FOIL gives us\( a^2 - 7a - 7a + 49 \).

Expanding Expressions

Once the multiplication is complete using methods like FOIL, the next step is to expand the expression. This process involves writing out the expression as a sum (or difference) of terms.

It makes the final expression clearer and ready for any further operations like simplification. In our example, after applying FOIL and writing out\( a^2 - 7a - 7a + 49 \), we have expanded the expression.

Expanding expressions is an essential skill in algebra, allowing students to work with complex polynomials comfortably. It sets the foundation for more advanced topics like polynomial division and factoring.

Binomial Squares

Binomial squares represent a common special case in algebra where a binomial is multiplied by itself. The general formula for squaring a binomial \((a - b)^2\) is:

• \((a-b)^2 = a^2 - 2ab + b^2\)

This formula arises naturally when two identical binomials are multiplied. It provides a quick way to find the square without manual distribution every time. In our exercise, applying this formula to\((a-7)\) yields:\(a^2 - 14a + 49\).

Understanding binomial squares is particularly useful when working with quadratic equations and expressions. It simplifies calculations and enhances problem-solving efficiency in many algebraic contexts.