Problem 40 - Elementary Algebra

Question

Factor the greatest common factor from each polynomial. \(-3 b+12\)

Step-by-Step Solution

Answer

3(-b + 4)

Step 1: Identify the terms in the polynomial

The polynomial given is -3 b + 12. The terms in the polynomial are -3b and 12.

Step 2: Find the greatest common factor (GCF)

To find the greatest common factor, determine the largest number that can divide both -3 and 12. The GCF of -3 and 12 is 3.

Step 3: Rewrite the polynomial with the GCF

Factor out the greatest common factor 3 from each term in the polynomial. This gives the expression as: -3 b + 12 = 3(-b + 4).

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into simpler components. It is like reverse distributing a product. In this process, we aim to express a polynomial as a product of its factors. For the polynomial -3b + 12, we want to find what common factors can simplify this expression. By factoring, we can solve polynomial equations more easily and simplify algebraic expressions for further operations. Remember, always check if there are common factors in all terms of the polynomial before proceeding with more complex factoring techniques.

GCF Identification

Identifying the Greatest Common Factor (GCF) is a necessary step in factoring polynomials. The GCF is the largest number that divides each term in the polynomial. To find the GCF of given terms:

• List the factors of each term.
• Find the common factors.
• Select the largest common factor.

For -3b and 12, the factors of -3 are ±1 and ±3, and the factors of 12 are ±1, ±2, ±3, ±4, ±6, and ±12. The common factors are ±1 and ±3. Hence, the GCF is 3. Once identified, you can factor the GCF out from each term in the polynomial.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables (like b), and operation symbols (+, -, ×, ÷). Simplifying these expressions can make solving equations easier. When given an expression like -3b + 12, it is important to recognize each part:

• -3b is a term with a variable and a coefficient (-3).
• 12 is a constant term.

By factoring out the GCF, we simplify it to 3(-b + 4). This smaller, more manageable expression can help in solving or further manipulating the algebraic equation. Always ensure to combine and reduce terms properly to avoid mistakes.