Problem 6 - Algebra 1: Concepts and Skills

Question

Copy the equation and fill in the blanks. \((x+2)(x+6)=x^{2}+\underline{?}+12\)

Step-by-Step Solution

Answer

The number that fits into the blank of the expression \(x^{2}+\underline{?}+12\) is 8.

Step 1: Apply the FOIL Method

First, apply the FOIL method to the expression on the left side of the equation. For the given expression, (x+2)(x+6), the FOIL method implies: First terms: \(x \cdot x = x^{2}\), Outer terms: \(x \cdot 6 = 6x\), Inner terms: \(2 \cdot x = 2x\), Last terms: \(2 \cdot 6 = 12\).

Step 2: Combine Like Terms

Next, combine like terms to simplify the expression from the previous step. The like terms are \(6x\) and \(2x\), which summation gives \(6x+2x=8x\). The expression would now look like \(x^{2}+8x+12\).

Step 3: Compare to the Given Equation

Finally, compare the expanded and simplified form \(x^{2}+8x+12\) with the given equation \(x^{2}+?+12\). Looking at both expressions, you should find that the value of '?' is \(8\).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one key process is combining like terms. This involves adding or subtracting variables that have the same power. For instance, in the step-by-step solution provided, the expression obtained from the FOIL method included two like terms: 6x and 2x. Because they are both terms in x to the first power, we can combine them by adding their coefficients to get 8x.

It's essential to remember that we can only combine terms that correspond to the same variable and exponent. Constants (numbers without variables) can be combined with other constants, while coefficients (numbers in front of variables) can be summed or subtracted if their variables and exponents match.

Binomial Multiplication

Binomial multiplication is a method used to multiply two binomials—polynomial expressions that each contain two terms. The textbook solution used the FOIL method for this task; FOIL stands for First, Outer, Inner, Last. It refers to the terms' position in each binomial.

First, you multiply the first terms in each binomial. Afterwards, multiply the outer terms, then the inner terms, and lastly the last terms of each binomial. In our exercise, multiplying the first terms x and x gave x^2, the outer terms x and 6 gave 6x, the inner terms 2 and x gave 2x, and lastly, the last terms 2 and 6 gave 12.

Polynomial Expressions

Polynomial expressions are algebraic statements that include variables raised to whole number exponents, and may also have constants and coefficients. A polynomial can have multiple terms, such as the trinomial in the FOIL example exercise: x^2 + 8x + 12.

Understanding the structure of polynomial expressions is crucial to performing operations like addition, subtraction, and multiplication on them. Each term in a polynomial is called a monomial, and when terms are combined through addition or subtraction, they form a larger polynomial. For example, a binomial has two terms, a trinomial has three terms, and the name changes with the number of terms. To deal with polynomials effectively, you should become comfortable with operations like the FOIL method, combining like terms, and further factorization and expansion techniques.