Problem 10 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ x^{2}+6 x+8 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial $$x^2 + 6x + 8$$ is $$(x + 2)(x + 4)$$.

Step 1: Identify the terms of the trinomial

The trinomial given is $$x^2 + 6x + 8$$. The terms are: - First term: $$x^2$$ - Middle term: $$6x$$ - Last term: $$8$$

Step 2: Break down the middle term

We need to find two numbers that multiply to give the last term (8) and add up to the coefficient of the middle term (6). The two numbers that fit this requirement are 2 and 4, since (2 × 4 = 8) and (2 + 4 = 6). So, we can break down the middle term (6x) into (2x + 4x).

Step 3: Rewrite the trinomial

Rewriting the trinomial by breaking down the middle term, we get: $$ x^2 + 2x + 4x + 8 $$

Step 4: Factor by grouping

We need to find the greatest common factor (GCF) of the first two terms and the last two terms, then factor them: - GCF of $$x^2$$ and $$2x$$ is $$x$$. So, taking $$x$$ as a common factor, we get: $$ x(x + 2) $$ - GCF of $$4x$$ and $$8$$ is $$4$$. So, taking $$4$$ as a common factor, we get: $$ 4(x + 2) $$ Now, we rewrite the trinomial combining these GCFs: $$ x(x + 2) + 4(x + 2) $$

Step 5: Factor the trinomial

We observe that both terms have a common factor of $$(x + 2)$$. So, we take $$(x + 2)$$ as a common factor and get: $$ (x + 2)(x + 4) $$ So, the factored form of the given trinomial $$x^2 + 6x + 8$$ is $$ (x + 2)(x + 4) $$

Key Concepts

Polynomials

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For our example, the expression \( x^2 + 6x + 8 \) is a quadratic trinomial, which means it contains three terms and its highest degree term is squared. This kind of expression forms the foundational building blocks for more complex algebraic operations.Polynomials are categorized based on their degree (the highest power of the variable) and the number of terms they contain:

• Monomials: Single-term expressions, like \( 3x^2 \).
• Binomials: Two-term expressions, like \( x - 2 \).
• Trinomials: Three-term expressions, like \( x^2 + 6x + 8 \).

Understanding the structure of polynomials helps in factoring them, a process often used to simplify algebraic expressions or solve equations.

Greatest Common Factor

The greatest common factor (GCF) is the highest number or variable term that divides two or more terms without leaving a remainder. When factoring polynomials, identifying the GCF is crucial as it simplifies the expression by taking out a common term from the groups of terms.In the trinomial \( x^2 + 6x + 8 \), when factoring by grouping we observe:

• For the terms \( x^2 \) and \( 2x \), the GCF is \( x \), giving us \( x(x + 2) \).
• For the terms \( 4x \) and \( 8 \), the GCF is \( 4 \), resulting in \( 4(x + 2) \).

Recognizing these common factors allows combining terms during the factoring process. Hence, understanding GCF aids in refining and solving algebraic expressions efficiently.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition and multiplication). They form the language through which most of algebra is expressed. The expression \( x^2 + 6x + 8 \) is an algebraic expression that requires understanding its components before factoring.Key aspects of algebraic expressions include:

• Variables: Symbols, like \( x \), that represent numbers.
• Coefficients: Numbers that multiply the variables, such as 6 in \( 6x \).
• Constants: Fixed numbers on their own, like 8 in the expression above.

Factoring is a technique used to simplify these expressions, often by reversing the process of expanding them. By breaking down and regrouping components, expressions can be transformed into a product of simpler factors, easing the path to finding their solutions or simplifying computations.