Problem 40 - Algebra A Combined Function

Question

Factor out the GCF from each polynomial. $$ x^{9} y^{6}+x^{3} y^{5}-x^{4} y^{3}+x^{3} y^{3} $$

Step-by-Step Solution

Answer

The GCF is \(x^3y^3\) and the factored form is \(x^3y^3(x^{6}y^{3} + y^{2} - xy + 1)\).

Step 1: Identify the GCF of the Coefficients and Variables

The coefficients of the terms are: 1 (for all terms since there are no numerical coefficients). The variables are in the form of powers of \(x\) and \(y\). The smallest power of \(x\) across all terms is \(x^3\) and for \(y\) is \(y^3\). Therefore, the GCF of the variables is \(x^3y^3\).

Step 2: Factor the GCF Out of Each Term

Factor \(x^3y^3\) out of each term of the polynomial: \[ x^3y^3(x^{6}y^{3} + y^{2} - xy + 1) \]

Step 3: Verify the Factored Form

To verify, distribute \(x^3y^3\) back into the terms to see if you get the original polynomial: \(x^{9}y^{6} + x^3y^5 - x^4y^3 + x^3y^3\). The expression matches, confirming the factorization is correct.

Key Concepts

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is an essential concept in algebra. It is the largest factor that divides each term in a polynomial. To find the GCF, you need to examine both coefficients and variables within each term.

• Coefficients: These are usually numbers, and you find the GCF by looking for the largest number that divides each coefficient in the polynomial. In our example, all coefficients are 1, so the GCF of the coefficients is also 1.
• Variables: When considering variables, you look at the lowest power of each variable present across all terms. For instance, if you are dealing with terms having powers like \(x^9, x^3, x^4\), the smallest power is \(x^3\). Similarly, for powers like \(y^6, y^5, y^3\), the smallest is \(y^3\). Therefore, the GCF of the variables is \(x^3y^3\).

Finding the GCF is the first step in factoring polynomials. It simplifies expressions and helps in solving algebraic equations more efficiently.

Polynomial Expressions

Polynomial expressions are mathematical expressions that involve sums of powers of variables with coefficients. Each term in a polynomial consists of a coefficient multiplied by a variable raised to a power. In our case, the polynomial is:\[x^9y^6 + x^3y^5 - x^4y^3 + x^3y^3\] There are several features of polynomial expressions to remember:

• Terms: Each part of the expression separated by plus or minus signs is called a term.
• Degree: The degree of a polynomial is the highest sum of the exponents of the variables in a single term. For the term \(x^9y^6\), the degree is \(9+6 = 15\).
• Variables and Exponents: In polynomials, variables can have non-negative integer powers, and these powers dictate the behavior of the polynomial graphically and algebraically.

Understanding polynomial expressions forms the foundation for more complex algebraic concepts and operations.

Algebraic Factoring

Algebraic factoring involves writing a polynomial as a product of its factors. It is an essential skill for solving polynomial equations, simplifying expressions, and finding roots. The process typically begins by identifying the greatest common factor, after which the polynomial is divided by this GCF:- Extract the GCF: Identify and extract the GCF from all terms. In our example, the GCF is \(x^3y^3\).- Factor the Polynomial: Rewrite the polynomial as a product of the GCF and another polynomial. The factored form of our example is:\[x^3y^3(x^6y^3 + y^2 - xy + 1)\]After factoring out the GCF, you'll often work with simpler polynomials, which can sometimes be factored further depending on their structure. Additionally, always verify your factored expression by expanding it back to ensure it matches the original expression.

Algebraic Expressions

Algebraic expressions include numbers, variables, and the operations of addition, subtraction, multiplication, and division. They are fundamental in mathematics, setting the groundwork for algebra. Each expression is a mathematical statement, forming the language through which algebra is communicated.

• Components: Just like sentences in language, algebraic expressions have parts that must be understood: terms, coefficients, and constants.
• Simplifying: Simplification is a common task in working with these expressions, often involving factoring, combining like terms, and performing arithmetic operations.
• Evaluating: Algebraic expressions can be evaluated by substituting values for the variables and performing the operations, yielding specific results.

Recognizing and manipulating algebraic expressions is crucial for solving equations, graphing functions, and applying algebra to real-world problems. The expression \(x^9y^6 + x^3y^5 - x^4y^3 + x^3y^3\) demonstrates the complexity that can arise in polynomial and algebraic expressions, requiring techniques like factorizations to simplify and solve.