Problem 12 - Elementary Algebra

Question

For the following problems, find the products. $$ (x+6)^{2} $$

Step-by-Step Solution

Answer

Answer: The product of \((x+6)^2\) is \(x^2 + 12x + 36\).

Step 1: Apply the square of a binomial formula

We can find the product of \((x+6)^2\) are using the square of a binomial formula: $$(A+B)^2 = A^2 + 2AB + B^2$$ Here, \(A = x\) and \(B = 6\). Let's substitute the values of A and B in the formula and simplify.

Step 2: Substitute the values of A and B

By substituting \(A = x\) and \(B = 6\), we have: $$(x+6)^2 = x^2 + 2(x)(6) + (6)^2$$

Step 3: Simplify the expression

Now we need to simplify the expression: $$(x+6)^2 = x^2 + 12x + 36$$ The product of \((x+6)^2\) is \(x^2 + 12x + 36\).

Key Concepts

Polynomial Multiplication

In algebra, one of the crucial techniques you'll learn is polynomial multiplication. This involves finding the product of two or more polynomials, which are expressions made up of variables and coefficients. For instance, when we talk about multiplying a polynomial like

• (x + 6)^2

we're essentially finding out what happens when you multiply two identical expressions together.
When multiplying polynomials, there are essential rules and formulas that assist in the process. A typical method is using the distributive property, which involves multiplying each term in one polynomial by all terms in the other polynomial.
However, specific polynomials have shortcuts; for instance, the square of a binomial formula, \[(A+B)^2 = A^2 + 2AB + B^2\] This formula allows you to efficiently calculate the square of a binomial without expanding it fully. It’s a time-saver when combined with strategic simplification.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. In the expression \((x + 6)^2\), you see two components: the variable

• x,

which can take on various values, and the constant

• 6,

which remains unchanged.
To work with these expressions, you should understand terms, coefficients, and like terms. A term is each element in the expression separated by a plus or minus sign. In \(x^2 + 12x + 36\), there are three terms:

• x^2,
• 12x, and
• 36.

The coefficients are the numerical parts of terms that accompany the variables—in this expression,

• 1 as the coefficient of x^2
• and
• 12 as the coefficient of x.

Understanding these components helps simplify and manipulate algebraic expressions more effectively.

Quadratic Expressions

Quadratic expressions are a specific type of algebraic expression that take the form \(ax^2 + bx + c\), where

• a,
• b, and
• c

are constants, and

• a ≠ 0.

These quadratics are fascinating because they have a degree of 2, signifying that the highest exponent of the variable is 2. When expanding \((x + 6)^2\), the result is \(x^2 + 12x + 36\), a perfect example of a quadratic expression.
Quadratics are the foundation of many algebraic applications, such as solving equations, exploring parabolic graphs, and more. Mastering how to identify and work with quadratic expressions can significantly improve your proficiency in handling more complex algebraic problems.