Problem 93 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (3-a)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is \(a^{2} - 6a + 9\).

Step 1: Recall the binomial expansion formula

The binomial expansion formula is given by: \((a+b)^{2} = a^{2} + 2ab + b^{2}\). Here, \(a = 3\) and \(b = -a\).

Step 2: Substitute the values of 'a' and 'b' into the formula

Now we will plug in the values of 'a' and 'b' into the formula: \((3-a)^{2} = (3)^{2} + 2(3)(-a) + (-a)^{2}\)

Step 3: Simplify the terms

We can simplify the expression as follows: \((3-a)^{2} = 9 - 6a + a^{2}\) So, the simplified algebraic expression is: \((3-a)^{2} = a^{2} - 6a + 9\).

Key Concepts

Binomial Expansion

The binomial expansion is a vital concept in algebra that helps us simplify expressions that involve powers of sums. The term "binomial" characterizes equations or expressions that contain two terms, much like \[(a + b)^n\].The binomial theorem provides a systematic way to expand these expressions. For a square, specifically two terms raised to the power of two, the formula simplifies to:\[(a+b)^2 = a^2 + 2ab + b^2\]In the example given, \[(3-a)^2\], the two terms are 3 and \(-a\). Plugging these into the formula gives:

• First term squared: \(3^2 = 9\)
• Twice the product of the terms: \(2 \times 3 \times (-a) = -6a\)
• Second term squared: \((-a)^2 = a^2\)

This ability to transform complex expressions using binomial expansion not only simplifies them but also sets the ground for efficient calculations and understanding of more complex algebra.

Simplification of Expressions

Simplification is a crucial skill when working with algebraic expressions. The goal is to present the equation in its simplest form while maintaining equality. In the given example, simplifying \( (3-a)^2 \) involves recognizing and correctly applying the binomial expansion formula.For simplification:

• Identify structure in the expression for possible formulas, like the binomial expansion.
• Systematically perform the expansion to transform the expression: \(9 - 6a + a^2\).
• Rearrange the terms to achieve the simplest version: \(a^2 - 6a + 9\).

Using simplification techniques, we ensure we have the simplest possible form of the expression which is easier to work with and interpret. Concepts such as combining like terms and recognizing symmetrical forms all assist in making algebra more manageable.

Polynomials

Polynomials are fundamental entities in mathematics comprised of variables raised to whole number powers, along with coefficients. A polynomial can have one variable or many and includes operations like addition, subtraction, and multiplication.The expression from our exercise, \(a^2 - 6a + 9\), is a polynomial, specifically a quadratic because the highest power of the variable \(a\) is 2.Key characteristics of polynomials:

• Terms: Individual parts separated by plus or minus signs, like \(a^2\), \(-6a\), and \(9\).
• Degree: Determined by the highest power of the variable, here it is 2 (quadratic).
• Coefficients: Numbers multiplied by the variables, such as \(-6\) in \(-6a\).

Understanding polynomials is essential because they often form the basis of much more complex mathematical contexts and their simplification is a common task in solving equations and modeling real-world situations. Mastering polynomials will not only help in academics but also in various fields like engineering, physics, and computer science.