Problem 42 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ 14 x^{3} y+7 x^{2} y-7 x y $$

Step-by-Step Solution

Answer

The GCF is \(7xy\); the factored form is \(7xy(2x^2 + x - 1)\).

Step 1: Identify the Terms

The given polynomial is \(14x^3y + 7x^2y - 7xy\). It consists of three terms: \(14x^3y\), \(7x^2y\), and \(-7xy\).

Step 2: Determine the Coefficients' GCF

The coefficients are 14, 7, and -7. The greatest common factor (GCF) of these coefficients is 7.

Step 3: Determine the Variable GCF

Each term has at least one \(x\) and one \(y\). So, the GCF for the variables is \(xy\).

Step 4: Combine the GCFs

Combine the GCFs found: the GCF of the coefficients is 7 and of the variables is \(xy\), so the overall GCF for the polynomial is \(7xy\).

Step 5: Factor out the GCF

Divide each term of the polynomial by the GCF \(7xy\) and factor it out:- \(\frac{14x^3y}{7xy} = 2x^2\)- \(\frac{7x^2y}{7xy} = x\)- \(\frac{-7xy}{7xy} = -1\)Therefore, the factored form is \(7xy(2x^2 + x - 1)\).

Key Concepts

greatest common factor (GCF)

In mathematics, the greatest common factor (GCF) is a valuable concept when simplifying expressions or solving equations. The GCF refers to the largest number that can evenly divide each of the given numbers of a polynomial without leaving a remainder. To find the GCF, follow these steps:

1. **List the Factors:** Write down all factors of each number.
2. **Identify the Common Factors:** Find the largest factors that appear in all lists.
3. **Determine the GCF:** The largest common factor from these lists becomes the GCF.

For example, in the polynomial expression \(14x^3y + 7x^2y - 7xy\), you first examine the numerical coefficients: 14, 7, and -7. Both 14 and 7 can be divided by the number 7, making it the GCF of these coefficients.

In algebraic expressions, identifying the GCF can make simplifying expressions easier, as seen when factoring polynomials. By extracting the GCF, you can rewrite the polynomial in a simpler form, making solving it more straightforward.

polynomial expressions

Polynomial expressions are algebraic expressions made up of variables and coefficients, connected by operations of addition, subtraction, and multiplication. Each separate part of a polynomial is called a "term," which can include constants and variables raised to a non-negative integer exponent. A polynomial's degree is determined by the highest exponent present.

For example, the polynomial expression \(14x^3y + 7x^2y - 7xy\) can be broken down as:

• Three distinct terms: \(14x^3y\), \(7x^2y\), and \(-7xy\).
• Each term includes coefficients (14, 7, and -7) and variables \(x\) and \(y\).

Understanding polynomial expressions is crucial in algebra since they are foundational for further topics like solving equations, factoring, and calculus. These expressions can be categorized based on the number of terms they contain:

• Monomial: A single term (e.g., \(7xy\)).
• Binomial: Two terms (e.g., \(x^2 + y\)).
• Trinomial: Three terms (e.g., \(x^2 + xy + y\)).

Factoring, especially through recognizing the GCF, helps in breaking down complex polynomial expressions into simpler parts, useful for calculations or solving equations.

algebraic terms

Algebraic terms are components of polynomial expressions that may include numbers, variables, or both, combined through multiplication. These terms are the building blocks of polynomials, with each term separated by a plus \(+\) or minus \(-\) sign.

In the expression \(14x^3y + 7x^2y - 7xy\), the algebraic terms are

• \(14x^3y\) - first term with 14 as the coefficient.
• \(7x^2y\) - second term with 7 as the coefficient.
• \(-7xy\) - third term with -7 as the coefficient. The minus sign is part of the term.

When simplifying expressions or factoring polynomials, it's essential to analyze these individual terms
by identifying common factors and degrees in each term. This helps in determining the greatest common factor (GCF) and rewriting the polynomial expression in a simpler, more manageable form.

Since each algebraic term consists of a coefficient and a variable part, understanding how to manipulate these elements is key for various algebraic operations. This includes distributing, factoring, and expanding factors in polynomial expressions.