Problem 39 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ a^{7} b^{6}-a^{3} b^{2}+a^{2} b^{5}-a^{2} b^{2} $$

Step-by-Step Solution

Answer

The GCF is \( a^2 b^2 \), factor it to get: \( a^2 b^2 (a^5 b^4 - a b + b^3 - 1) \).

Step 1: Determine the GCF of coefficients

Each term in the polynomial has a leading coefficient of 1, so there's no numerical greatest common factor (GCF) for the coefficients other than 1.

Step 2: Determine the GCF of the variable 'a'

The exponents of 'a' in each term are 7, 3, 2, and 2 respectively. The smallest exponent among these is 2, so the GCF for 'a' is \( a^2 \).

Step 3: Determine the GCF of the variable 'b'

The exponents of 'b' in each term are 6, 2, 5, and 2 respectively. The smallest exponent among these is 2, so the GCF for 'b' is \( b^2 \).

Step 4: Combine the GCF of variables

Combine the GCF of 'a' and 'b' to form the overall GCF of the polynomial, which is \( a^2 b^2 \).

Step 5: Factor out the GCF

Factor out \( a^2 b^2 \) from each term in the polynomial. \[ a^{7} b^{6} - a^{3} b^{2} + a^{2} b^{5} - a^{2} b^{2} = a^2 b^2 (a^5 b^4 - a b + b^3 - 1) \]

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor (GCF) in mathematics is the largest number or expression that divides two or more numbers or expressions without leaving a remainder. When dealing with polynomials, the GCF refers to the highest degree of similarity among the terms of the polynomial.
When we say the GCF of a polynomial, we are looking at:

• The numbers (or coefficients) in the polynomial.
• The variables with the smallest power in all terms.

Consider a term like \(a^7 b^6\) and another like \(a^3 b^2\); the common part is \(a^3 b^2\). Factors are expressions multiplied together to get another expression. Finding the GCF helps simplify expressions, making them easier to solve or manipulate.
In practice, finding the GCF involves:- Looking for common numerical factors in coefficients.- Identifying variables raised to the smallest powers across the terms, as these indicate how many times that variable can evenly divide each term.
By factoring out the GCF, we essentially "pull out" the biggest common piece these terms share, simplifying the polynomial for further operations like solving or graphing.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication). They are fundamental in expressing mathematical relationships and solving equations.
Algebraic expressions can be as simple as \(a + b\) or more complex like \(a^2 b^2 + 3ab - 4\). They can include:

• Constants, which have a fixed value, like 3 or -4.
• Variables, which represent unknown or changeable values.
• Coefficients, which are numbers that multiply the variables.
• Exponents, showing how many times a variable is multiplied by itself.

A polynomial is a type of algebraic expression with one or more terms, and each term can be a combination of constants, variables, and exponents.
Factoring uses algebraic expressions to break down a polynomial into simpler parts (or factors), helping in functions like deriving and integrating, solving equations, and simplifying expressions.

Variable Exponents

Variable exponents are a key part of polynomials and algebraic expressions, indicating how many times a variable is used as a factor. In the expression \(a^7\), the 7 is the exponent, telling us that the variable 'a' is multiplied by itself seven times.
When dealing with polynomials like \(a^{7} b^{6} - a^{3} b^{2} + a^{2} b^{5} - a^{2} b^{2}\), each term's exponents help determine our GCF.
To find the GCF related to variable exponents:

• List the exponents of the same variable from each term.
• Choose the smallest exponent among them for that particular variable.

The reasoning is similar to numbers: if you have repeated factors, the GCF uses the lowest count across terms, ensuring it divides each term in the polynomial evenly.
Understanding variable exponents is crucial for organizing and simplifying expressions, as they guide how terms interrelate and transform within algebraic manipulations.