Problem 41 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(21 b^{2}-48 b-45\)

Step-by-Step Solution

Answer

The trinomial \(21b^2 - 48b - 45\) factors to \(3(b - 3)(7b + 5)\).

Step 1: Identify and Write Down the Trinomial

The given trinomial is \(21b^2 - 48b - 45\). This is a quadratic trinomial, where \(a = 21\), \(b = -48\), and \(c = -45\).

Step 2: Find the Product of 'a' and 'c'

Calculate the product of the coefficients \(a\) and \(c\), which are \(21\) and \(-45\) respectively. The product is \[ 21 imes (-45) = -945 \].

Step 3: Find Two Numbers That Multiply to '-945' and Add to '-48'

Identify two numbers whose product is \(-945\) and sum is \(-48\). After checking factors, the numbers \(15\) and \(-63\) satisfy this requirement because \(15 + (-63) = -48\) and \(15 imes -63 = -945\).

Step 4: Rewrite the Middle Term Using the Two Numbers

Rewrite the quadratic trinomial by splitting the middle term \(-48b\) into \(15b - 63b\). Thus, the expression becomes \[ 21b^2 + 15b - 63b - 45 \].

Step 5: Factor by Grouping

Group the terms into pairs:\[ (21b^2 + 15b) + (-63b - 45) \].Factor out the greatest common factor from each pair:1. From \(21b^2 + 15b\), factor \(3b\), giving \(3b(7b + 5)\).2. From \(-63b - 45\), factor \(-9\), giving \(-9(7b + 5)\).

Step 6: Complete the Factorization

Since both terms contain \((7b + 5)\), factor \((7b + 5)\) out:\[ 3b(7b + 5) - 9(7b + 5) = (3b - 9)(7b + 5) \].

Step 7: Simplify the Expression

Further factor \(3b - 9\) by finding the greatest common factor, which is 3.Thus, \(3b - 9 = 3(b - 3)\).Combine to get the fully factored form:\[ 3(b - 3)(7b + 5) \].

Key Concepts

Quadratic Trinomials

A quadratic trinomial is a polynomial with three terms, each powered by a non-negative integer, with the highest degree being two. In the form \(ax^2 + bx + c\), these trinomials have specific characteristics that make them identifiable in algebra. Understanding quadratic trinomials involves recognizing their terms:

• The first term \(ax^2\) often sets the stage for further calculations with the leading coefficient \(a\).
• The middle term, \(bx\), influences the polynomial's behavior and symmetry.
• The constant term \(c\) closes out this trio.

To solve quadratic trinomials, recognizing these parts helps in the factorization process, which transforms the expression into a product of simpler polynomials. This enables easier solving of equations or understanding of polynomial behavior.

Factorization

Factorization is a key algebraic method used to break down polynomials into simpler "building blocks" or factors. These factors, when multiplied, give the original polynomial. In simpler terms, factorization splits a complex expression into a product of two or more simpler terms. The process often involves:

• Finding pairs of numbers whose product equals the given terms, a technique known as the "AC method" for quadratic equations.
• Using techniques such as splitting the middle term for more complex quadratic trinomials.
• Applying grouping methods to simplify expressions.

Factorization not only aids in solving polynomial equations but also in simplifying expressions for further mathematical operations.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operations like addition and multiplication. These are the language of algebra, and mastering them is crucial for mathematical fluency. In the context of factoring polynomials, algebraic expressions can be categorized into different types, such as monomials, binomials, and polynomials like the quadratic trinomial.

• The use of expressions allows us to represent unknown values and relationships succinctly.
• Factorization transforms these complex expressions into simpler, more manageable forms.

Understanding how to manipulate algebraic expressions through operations such as addition, subtraction, and particularly factorization, is an essential skill in algebra.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by two or more terms. It plays a pivotal role in simplifying algebraic expressions and is the first step in many factorization processes. To determine the GCF, one must look at:

• The numerical coefficients, finding the largest integer that divides all.
• Any common variable parts, determining the lowest power that appears in each term.

For instance, in the expression from the original exercise, finding the GCF for each group helped simplify the factorization process. This simplification makes complex algebraic problems much more manageable, allowing for straightforward calculations and clearer insights into the expression's structure.