Problem 63 - Elementary Algebra

Question

(a) factor by grouping. Identify any prime polynomials. (b) check. $$ 3 a c+b d-3 a d-b c $$

Step-by-Step Solution

Answer

The polynomial factors to \( (3a - b)(c - d) \).

Step 1: Group the terms

Group the terms in pairs to make the factoring process easier: \(3ac + bd - 3ad - bc\).

Step 2: Factor out the common factor from each group

Factor out the common factor from each group: \(3ac - 3ad\) and \(bd - bc\). This results in: \( 3a(c - d) + b(d - c) \).

Step 3: Factor out the common binomial factor

Notice that \( c - d \) and \( d - c \) are opposites. Rewrite \( b(d - c) \) as \( -b(c - d) \), and then factor out the common binomial factor \( c - d \): \( 3a(c - d) - b(c - d) \).

Step 4: Simplify the expression

Factor \( c - d \) from the expression: \( (3a - b)(c - d) \).

Step 5: Check the factorization

Expand \( (3a - b)(c - d) \) to confirm it matches the original polynomial: \( (3a - b)(c - d) = 3ac - 3ad - bc + bd \). The factorization is correct.

Key Concepts

polynomial factorization

Polynomial factorization involves breaking down a polynomial into simpler factors. These factors multiply to give the original polynomial.
In the problem, we have the polynomial expression: \(3ac + bd - 3ad - bc\).
The goal is to factor it into simpler expressions. We start by grouping the terms.

common factors

Identifying common factors in a polynomial helps break it down into simpler parts. In the polynomial expression given, we group terms to find common factors:
\(3ac - 3ad\) and \(bd - bc\).
From the first pair, the common factor is \(3a\).
From the second pair, the common factor is \(b\).
Factoring these out, we get: \(3a(c - d) + b(d - c)\).

binomial expressions

A binomial expression contains two terms. Here, the binomial expressions \((c - d)\) and\((d - c)\) appear after factoring out the common factors from each group:
\(3a(c - d) + b(d - c)\).
Notice that \(d - c\) is the opposite of \(c - d\).
By rewriting \(b(d - c)\) as \(-b(c - d)\), we can factor out the common binomial expression \((c - d)\):
\(3a(c - d) - b(c - d)\)
which simplifies to \((3a - b)(c - d)\).

elementary algebra

Elementary algebra involves basic algebraic operations, including factorization, which we used in this exercise. The steps taken are fundamental techniques:

• Grouping terms to simplify the expression.
• Factoring out common factors.
• Identifying binomial expressions and common factors within them.

We verified the factorization by expanding the simplified expression: \((3a - b)(c - d) = 3ac - 3ad - bc + bd\).
This matches the original polynomial, confirming our factorization is correct.