Problem 50 - Elementary Algebra

Question

For the following problems, find the products. $$ (x+6)(x-6) $$

Step-by-Step Solution

Answer

Answer: The product of the binomials (x+6) and (x-6) is $$x^2 - 36$$.

Step 1: First terms

Multiply the first terms in each binomial together, which are x and x. This gives: $$ x^2 $$

Step 2: Outer terms

Multiply the outer terms in the binomials, which are x and (-6). This gives: $$ -x^2(x+6)-6x $$

Step 3: Inner terms

Multiply the inner terms in the binomials, which are 6 and x. This gives: $$ 6x $$

Step 4: Last terms

Multiply the last terms in each binomial together, which are 6 and (-6). This gives: $$ -36 $$

Step 5: Combine and simplify

Combine all the terms from steps 1-4 and simplify: $$ x^2 - 6x + 6x - 36 $$ The terms "-6x" and "6x" cancel out, so the final simplified polynomial is: $$ x^2 - 36 $$ So the product of (x+6)(x-6) is $$x^2 - 36$$.

Key Concepts

Polynomial Multiplication

When faced with an algebraic task such as finding the product of two binomials, like \( (x+6)(x-6) \), we are dealing with the concept of polynomial multiplication. Polynomials are algebraic expressions that consist of variables and coefficients, connected by the operations of addition, subtraction, and multiplication.

For polynomial multiplication, each term in the first polynomial must be multiplied by each term in the second polynomial. The process follows a systematic approach often referred to as the FOIL method - First, Outer, Inner, and Last, representing the sequence of multiplying respective terms from each binomial.

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

Once these products are obtained, the next step is to combine like terms, which often involves adding or subtracting the products and simplifying to the most reduced form of the polynomial.

Difference of Squares

An interesting outcome of polynomial multiplication is revealed when multiplying certain types of binomials: the difference of squares. This occurs when multiplying two binomials that are identical except for the second term being the negative of the first, such as the provided example \( (x+6)(x-6) \).

The formula for the difference of squares is \( a^2 - b^2 = (a+b)(a-b)\). Applying this formula to our example, we set \(a=x\) and \(b=6\), leading to the simplified result \(x^2 - 36\), skipping much of the multiplication steps.

The reason the middle terms (\(6x\) and \( -6x\) in our example) cancel each other out is that they are equal in magnitude but opposite in sign. This concept is a powerful tool and can significantly shorten the multiplication process, especially with more complex polynomials or larger numbers.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. The expressions we have been working with, such as \( x+6 \) and \( x-6 \) from the exercise, exemplify algebraic expressions.

To deeply understand algebraic expressions, one must be familiar with terms and coefficients. A term is a single mathematical entity that can be a number, a variable, or both multiplied together. In the expression \( x^2 - 36 \), \( x^2 \) and \( -36 \) are considered terms, with \( x^2 \) having a coefficient of 1 (though typically it's not written).

Grasping the nature of algebraic expressions is crucial as they represent the foundation for much of algebra, including equations, inequalities, and functions. As we manipulate these expressions in operations like polynomial multiplication, we often aim to simplify or rearrange them to reveal patterns or to solve for specific variables.