Problem 26 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ a^{2}-3 a+2 $$

Step-by-Step Solution

Answer

Answer: $(a - 1)(a - 2)$

Step 1: Identify the coefficient and constant term

Coefficient of the linear term: -3 Constant term: 2

Step 2: Find two numbers that multiply to the constant term and add up to the coefficient of the linear term

In this case, the two numbers are -1 and -2, since (-1) * (-2) = 2 and (-1) + (-2) = -3.

Step 3: Rewrite the trinomial as a product of two binomials using the two numbers found in Step 2

We can rewrite the given trinomial using the two numbers we found, -1 and -2, as follows: $$a^2 - 3a + 2 = (a - 1)(a - 2)$$

Step 4: Present the final factored form of the given trinomial

The factored form of the given trinomial is: $$(a - 1)(a - 2)$$

Key Concepts

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients. They are constructed using operations like addition, subtraction, multiplication, and non-negative integer exponents of variables. The structure of polynomials makes them quite versatile in solving mathematical problems. A simple form of a polynomial is written as:\[ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]Here, \(a_n, a_{n-1}, \ldots, a_0\) are coefficients, and the variable \(x\) is typically used to represent the variable part.

Polynomials are classified based on their degree and the number of terms they contain:

• **Monomial:** A polynomial with a single term, like \(3x^2\).
• **Binomial:** A two-term polynomial, such as \(x + 2\).
• **Trinomial:** A polynomial with three terms, for example, \(x^2 - 3x + 2\).

Understanding polynomials helps in various mathematical operations, including factoring, which is essential for simplifying expressions and solving equations.

Binomials

A binomial is a polynomial that consists specifically of two terms. Each term contains a variable raised to a power and may be multiplied by a coefficient. A clear example is the expression \(a - 1\) or \(a + 2\).

In factoring trinomials, like the given example \(a^2 - 3a + 2\), the goal often is to express the trinomial as a product of two binomials. This involves finding two numbers that multiply to the constant term of the trinomial and add up to the coefficient of the linear term. For instance, in the trinomial \(a^2 - 3a + 2\), it can be factored into the binomials \((a - 1)(a - 2)\).

Working with binomials is crucial because they simplify expressions and make it easier to solve equations, especially in algebra and calculus.

Coefficient and Constant Terms

In any polynomial, including trinomials, terms are composed of two key elements: coefficients and constants. Understanding these components is vital for performing operations like factoring.

• **Coefficient:** A number multiplying a variable. In the expression \(-3a\), \(-3\) is the coefficient of \(a\).
• **Constant Term:** A number with no variable attached, showing its value is constant. For instance, in the expression \(+2\), \(2\) is a constant term.

Finding the right combination of coefficients and constant terms is a strategic way to factor polynomials. When factoring a trinomial, such as \(a^2 - 3a + 2\), we seek numbers whose product equals the constant term and whose sum equals the linear coefficient. In this case, these numbers are \(-1\) and \(-2\), leading to the factored form \((a - 1)(a - 2)\). Recognizing coefficients and constants swiftly is a foundational skill for more advanced algebraic manipulations.