Problem 34 - Basic Engineering Mathematics

Question

Factorize (a) \(x y-3 x z\) (b) \(4 a^{2}+16 a b^{3}\) (c) \(3 a^{2} b-6 a b^{2}+15 a b\)

Step-by-Step Solution

Answer

(a) \(x(y - 3z)\), (b) \(4a(a + 4b^3)\), (c) \(3ab(a - 2b + 5)\).

Step 1: Identify the Common Factor in (a)

For the expression \(xy - 3xz\), identify the common factor for the terms. Both terms have a factor of \(x\) that can be factored out.

Step 2: Factor Out the Common Factor in (a)

Factor \(x\) out from the expression:\[x(y - 3z)\]This simplifies it to \(x(y - 3z)\).

Step 3: Identify the Common Factor in (b)

For the expression \(4a^2 + 16ab^3\), identify the greatest common factor (GCF). The GCF here is \(4a\), as both terms are divisible by this.

Step 4: Factor Out the Common Factor in (b)

Factor \(4a\) out of the expression:\[4a(a + 4b^3)\]The expression \(4a^2 + 16ab^3\) factorizes to \(4a(a + 4b^3)\).

Step 5: Identify the Common Factor in (c)

For the expression \(3a^2b - 6ab^2 + 15ab\), the greatest common factor for all terms is \(3ab\).

Step 6: Factor Out the Common Factor in (c)

Factor \(3ab\) out of each term:\[3ab(a - 2b + 5)\]So the expression \(3a^2b - 6ab^2 + 15ab\) simplifies to \(3ab(a - 2b + 5)\).

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that can evenly divide two or more numbers or terms. Finding the GCF is a crucial step in the process of factorization. For algebraic expressions, the GCF helps in simplifying expressions by identifying the terms that can be factored out easily.
To find the GCF of multiple terms, consider the following steps:

• List the factors of each term individually.
• Highlight the common factors among the terms.
• Select the largest factor that appears in all lists as the GCF.

Applying this to algebraic terms involves both numerical coefficients and variables with their respective powers. For example, in the expression \(4a^2 + 16ab^3\), the GCF \(4a\) is both a number that divides the coefficients and a variable factor common in both terms. Once identified, the GCF can be factored out, leading to a simplified expression.

Mathematical Expressions

Mathematical expressions display relationships or operations performed on numbers, variables, or a combination of both. These expressions do not contain an equals sign, differentiating them from equations. Factorization is a common procedure applied to expressions to break them down into simpler components.
Expressions can include components such as:

• Variables (e.g., \(x, y\))
• Coefficients (the numbers in front of the variables)
• Operators (like \(+, -\))

Understanding the structure and components of a mathematical expression can make processes like factorization more intuitive. Recognizing patterns or common factors in expressions is key to applying techniques effectively. In the problem \(3a^2b - 6ab^2 + 15ab\), recognizing terms and their constituents helps uncover the greatest common factor for simplification.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in a simpler or more efficient form while maintaining their original value. This is often achieved by factoring, which helps to find the most reduced version of an expression. By simplifying expressions, calculations become more manageable.
In the context of the given problems, here’s how simplification works:

• Identify the greatest common factor across the terms in the expression.
• Divide each term by this GCF to confirm that it can be universally factored out.
• Express the original terms as a product of the common factor and a simplified expression.

Consider the expression \(xy - 3xz\). By factoring out the common \(x\), it reduces to \(x(y - 3z)\). This not only simplifies the expression but can also help in solving equations where this expression might be a part. Simplification lays the groundwork for more complex problem-solving in algebra.