Problem 54 - Elementary Algebra

Question

Factor out the GCF. $$ 4 x 5 y 4 z 9+26 x 5 y 3 z 4-14 x 6 y 8 z 5 $$

Step-by-Step Solution

Answer

The factored expression is \(2x^5y^3z^4(2yz^5 + 13 - 7xy^5z)\).

Step 1: Break Down Each Term

The given expression is \(4x^5y^4z^9 + 26x^5y^3z^4 - 14x^6y^8z^5\). We need to analyze each term separately:1. \(4x^5y^4z^9\) has coefficients and powers: 4 for the coefficient, 5 for \(x\), 4 for \(y\), and 9 for \(z\).2. \(26x^5y^3z^4\) has coefficients and powers: 26 for the coefficient, 5 for \(x\), 3 for \(y\), and 4 for \(z\).3. \(-14x^6y^8z^5\) has coefficients and powers: -14 for the coefficient, 6 for \(x\), 8 for \(y\), and 5 for \(z\).

Step 2: Identify the GCF of the Coefficients

Find the greatest common factor (GCF) of the numerical coefficients 4, 26, and 14. The prime factorizations are:1. 4: \(2^2\)2. 26: \(2 \, \times \, 13\)3. 14: \(2 \, \times \, 7\)The common prime factor is 2. Thus, the GCF of the coefficients is 2.

Step 3: Identify the GCF of the Variables

For each variable, determine the smallest power present in all terms:- \(x\): The smallest power is 5 (common to the first and second terms).- \(y\): The smallest power is 3 (from the second term).- \(z\): The smallest power is 4 (from the second term).Therefore, the GCF for the variables is \(x^5y^3z^4\).

Step 4: Combine GCF Components

Combine the GCF of the coefficients and the variables. The complete GCF is \(2x^5y^3z^4\).

Step 5: Factor Out the GCF

Divide each term in the expression by the GCF \(2x^5y^3z^4\) and factor it out:- \(4x^5y^4z^9 / 2x^5y^3z^4 = 2yz^5\)- \(26x^5y^3z^4 / 2x^5y^3z^4 = 13\)- \(-14x^6y^8z^5 / 2x^5y^3z^4 = -7xy^5z\)Thus, the expression becomes:\[2x^5y^3z^4(2yz^5 + 13 - 7xy^5z)\]

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and arithmetic operations such as addition, subtraction, multiplication, and division. It can include coefficients (numerical values multiplying variables), exponents (indicating the power to which a variable is raised), and several terms. Terms in an algebraic expression are separated by plus or minus signs, forming segments like \(4x^5y^4z^9\), \(26x^5y^3z^4\), or \(-14x^6y^8z^5\).
Understanding these parts of an algebraic expression helps in various operations including simplification, evaluation, and factoring.

• Each term has coefficients: the numerical part. For example, 4 in \(4x^5y^4z^9\).
• Variables like \(x, y, z\) can take different values. They are often raised to a power or exponent.
• An expression can have many terms, and identifying these correctly is crucial for any algebraic manipulation.

Recognizing the structure of algebraic expressions is foundational in algebra, allowing for the simplification of complex problems and understanding deeper concepts.

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler parts or factors that, when multiplied together, give back the original polynomial. This method simplifies expressions and solves equations more easily. When given an expression like \(4x^5y^4z^9 + 26x^5y^3z^4 - 14x^6y^8z^5\), factoring makes it more manageable.
In polynomial factoring, specifically when finding a greatest common factor (GCF), one seeks the highest common element in terms of coefficients and variables. This approach collapses multiple terms into a singular simpler expression or factor.

• Identify terms and separate them based on similarity or common factors.
• Seek common variables with the smallest power among given terms.

Factoring simplifies many algebraic processes, including solving equations and simplifying integrals or derivatives, proving crucial in countless mathematical applications.

Greatest Common Factor

The Greatest Common Factor, or GCF, of a set of terms is the largest polynomial or factor that divides each term in the set without leaving a remainder. Calculating the GCF involves looking at numerical coefficients and variable parts separately. It helps simplify polynomial expressions by factoring out the common element.
Consider the coefficients 4, 26, and 14 from the expression. Find the common factor through prime factorization. This yielded a GCF of 2.

• For variables like \(x, y, z\), the lowest power common in each variable determines the GCF part for variables.
• In this solution, the GCF for variables was \(x^5y^3z^4\).

Combining both, the full GCF of the given expression is \(2x^5y^3z^4\). Extracting or factoring this from the complete expression simplifies it considerably, making further mathematical manipulation or when solving equations straightforward.