Problem 90 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(t^{2}-3 t-54\)

Step-by-Step Solution

Answer

The factored form is \( (t + 6)(t - 9) \).

Step 1: Identify coefficients

For the trinomial of the form \(x^2 + bx + c\), identify the coefficients. Here, \(a = 1\), \(b = -3\), and \(c = -54\).

Step 2: Find two numbers that multiply to c

Find two numbers that multiply to \(c = -54\) and add up to \(b = -3\). These numbers are \(6\) and \(-9\) because \(6 \times (-9) = -54\) and \(6 + (-9) = -3\).

Step 3: Split the middle term

Rewrite \(-3t\) as \(6t - 9t\). The equation becomes \(t^2 + 6t - 9t - 54\).

Step 4: Factor by grouping

Group the terms to factor by grouping: \((t^2 + 6t) + (-9t - 54)\).

Step 5: Factor out common terms

Factor out the common terms in each group: \(t(t + 6) - 9(t + 6)\).

Step 6: Factor out the binomial

Since \(t + 6\) is common in both terms, factor it out: \((t + 6)(t - 9)\).

Key Concepts

Trinomial Factorization

Trinomial factorization involves breaking down a polynomial expression of the form x^2 + bx + cto a product of two binomials. This process makes solving equations and finding roots easier. For example, the trinomial t^2 - 3t - 54can be factored into (t + 6)(t - 9)through a set of clear steps. These steps often include finding two numbers that multiply to c and add to b, splitting the middle term, and then factoring by grouping. The goal is to rewrite the polynomial in a simpler form, facilitating further processing or solving.

Polynomial Expressions

Polynomial expressions consist of terms that are a combination of variables and coefficients. These terms are raised to non-negative integer powers and added or subtracted. For example, in the expression
t^2 - 3t - 54
the polynomial consists of three terms:

• t^2 (the quadratic term)
• -3t (the linear term)
• -54 (the constant term)

Understanding the structure of polynomials is essential, as it allows us to identify the coefficients and powers that determine the behavior of the polynomial. Polynomials can represent a wide variety of real-world problems and thus, mastering how to manipulate them, including factorization, is highly useful in both academic and practical contexts.

Factoring by Grouping

Factoring by grouping is a crucial technique in trinomial factorization. It involves splitting the middle term of a trinomial into two terms that can be grouped and factored separately. Here's how it works:
1. Split the middle term: Take the equation t^2 - 3t - 54Step 1: Find two numbers that multiply to -54 and add to -3. These numbers are 6 and -9.
Step 2: Rewrite the equation: t^2 + 6t - 9t - 54
2. Group the terms: (t^2 + 6t) + (-9t - 54)
3. Factor out the common terms in each group: t(t + 6) - 9(t + 6)
4. Factor out the binomial that's common: (t + 6)(t - 9)
This method helps simplify complex polynomial expressions and makes it easier to solve for variable values or simplify the equation further. Factoring by grouping can be especially handy for polynomials that cannot be easily factored by inspection.