Problem 123 - Algebra and Trigonometry Real Mathematics, Real People

Question

Factor the trinomial. $$x^{2}-6 x-27$$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(x^2 - 6x - 27\) is \((x - 9)(x + 3)\).

Step 1: Identify the Coefficients and Constant

The trinomial given is \(x^2 - 6x - 27\). The coefficients of the x terms are 1 (coefficients of x²) and -6 (coefficient of x), and the constant is -27.

Step 2: Find the Factors

Find two numbers that multiply to -27 and add up to -6. After examining the factors of 27, we see that -9 and 3 meet these conditions. Because -9 times 3 equals -27, and -9 plus 3 equals -6.

Step 3: Write the Factors

Now we write the trinomial as two binomial factors using the numbers we found. These factors will have the form (x - a)(x - b), where a and b are the numbers which we found. In this case, our factors are -9 and 3, so our two binomial factors are (x - 9) and (x + 3).

Key Concepts

Coefficients in Trinomials

Coefficients are the numerical values placed in front of the variables in an algebraic expression. In any trinomial of the form \( ax^2 + bx + c \), the number \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) represents the constant term. They play a crucial role in factoring trinomials because they define how the terms in the expression interact with each other.
Understanding coefficients is the first step in approaching a trinomial for factoring. For example, in the expression \( x^2 - 6x - 27 \), the coefficients and constant are as follows:

• Coefficient of \( x^2 \) is 1. This tells us that if we factor the trinomial completely, the leading term in our factors must multiply to \( x^2 \).
• Coefficient of \( x \) is -6. This is important to know for the middle term which represents the sum of the products of outer and inner terms when the trinomial is factored into binomials.
• Constant term is -27. This constant determines the terms in the binomial factors when set as a product to equal -27.

By understanding these coefficients, it becomes easier to identify suitable pairs of numbers that multiply to create the predetermined operation of the trinomial.

Understanding Binomial Factors

Binomial factors are expressions that contain two terms and are key when breaking down a trinomial into a simpler form. Each trinomial expression can often be represented as a product of two binomial factors, with the general structure \( (x - a)(x - b) \), where \( a \) and \( b \) are numbers derived from analyzing the trinomial's structure.
To find these binomial factors, you need to look at the constant term and the coefficient of \( x \). You want to identify two numbers that not only multiply to give the constant term but also add up to give the middle term's coefficient.
For \( x^2 - 6x - 27 \), the goal is to find numbers that fit these criteria:

• Product: -27 (the constant term)
• Sum: -6 (the coefficient of \( x \))

In this trinomial, the numbers -9 and 3 meet these conditions because \(-9 \times 3 = -27\) and \(-9 + 3 = -6\). Thus, the binomial factors are \( (x - 9) \) and \( (x + 3) \). Knowing how to determine these supporting figures helps simplify complex algebraic problems.

Multiplying Factored Binomials

Once binomial factors are determined, it's important to grasp how these multiply back into the original trinomial. This understanding verifies your factorization process. The core of this multiplication process is the distributive property, commonly referenced through the FOIL method in algebra, which stands for First, Outer, Inner, and Last.
When multiplying \( (x - 9)(x + 3) \):

• First: Multiply the first terms: \( x \times x = x^2 \)
• Outer: Multiply the outer terms: \( x \times 3 = 3x \)
• Inner: Multiply the inner terms: \(-9 \times x = -9x \)
• Last: Multiply the last terms: \(-9 \times 3 = -27 \)

Add these together to get \( x^2 + 3x - 9x - 27 \), which simplifies to \( x^2 - 6x - 27 \). Practice ensuring that these steps solidify your understanding of how binomials transform into trinomials, thus confirming that the factorization process was done correctly. These steps are crucial in verifying correctness in polynomial equations.