Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 16 x^{5} y^{2}+8 x^{3} y^{3} $$

Step-by-Step Solution

Answer

The factored form is \(8x^3y^2(2x^2 + y)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, we need to identify the greatest common factor (GCF) of the terms in the polynomial. The terms are \(16x^5y^2\) and \(8x^3y^3\). The numerical GCF of 16 and 8 is 8. For the variable part, the smallest power of \(x\) is \(x^3\) and the smallest power of \(y\) is \(y^2\). Thus, the GCF is \(8x^3y^2\).

Step 2: Factor Out the GCF

Next, factor the GCF, \(8x^3y^2\), out of each term. Divide each term by the GCF and rewrite the polynomial as a product: \[ \begin{align*} 16x^5y^2 + 8x^3y^3 &= 8x^3y^2(\frac{16x^5y^2}{8x^3y^2} + \frac{8x^3y^3}{8x^3y^2}) \ &= 8x^3y^2(2x^2 + y). \end{align*} \]

Step 3: Verify the Result

Finally, verify the factorization is correct by distributing \(8x^3y^2\) back into \((2x^2 + y)\): \(8x^3y^2 \cdot 2x^2 + 8x^3y^2 \cdot y = 16x^5y^2 + 8x^3y^3\). This shows the polynomial was correctly factored.

Key Concepts

Polynomials

A polynomial is a type of algebraic expression that consists of variables and coefficients. It's structured in terms of powers of these variables. In simple terms, a polynomial is a mathematical expression involving a sum of powers in one or more variables, each multiplied by a coefficient.
A polynomial can have constants, variables, and exponents. Here are the main components of polynomials:

• Terms: Individual components, like \(16x^5y^2\) or \(8x^3y^3\).
• Coefficients: The numerical part of each term. In \(16x^5y^2\), the coefficient is 16.
• Exponents: Indicate how many times the variable is multiplied by itself, such as 5 in \(x^5\).
• Variables: Symbols representing numbers we don't know yet, like \(x\) and \(y\).
• Degree: The highest sum of the exponents of variables in a single term. In \(16x^5y^2\), the degree is 7, as it combines powers of \(x\) and \(y\).

Understanding polynomials is crucial in algebra as they form the foundation for more complex expressions and equations.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers or algebraic expressions. It is a useful concept when simplifying expressions, particularly in factoring polynomials.
To find the GCF of algebraic terms:

• Find the numerical GCF: Identify the highest number that can evenly divide the coefficients of the terms. For instance, in 16 and 8, the GCF is 8.
• Identify the variables' lowest powers: Look at each variable separately and choose the smallest power present in all terms. In the polynomial \(16x^5y^2 \) and \(8x^3y^3\), the smallest power of \(x\) is \(x^3\), and the smallest power of \(y\) is \(y^2\). This makes the GCF \(8x^3y^2\).

Once identified, the GCF can be factored out from the polynomial to simplify it, making complex expressions manageable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition and subtraction). They are the building blocks in algebra. These expressions don't have an equals sign, setting them apart from equations, which do have one.
Key elements of algebraic expressions include:

• Variables: Represent unknown values and are usually denoted by letters such as \(x\) or \(a\).
• Constants: Fixed values in the expression such as numbers 3, 5, or 100.
• Operators: Symbols that indicate the operation to be performed, including +, -, \(*\), and \(/\).

Algebraic expressions allow us to model real-life situations with math, analyze patterns, and find relationships between different quantities. They can become complex, but understanding basic components helps to simplify, manipulate, and solve expressions, making challenges like factorization much easier.
By learning to work with these expressions, you form the basis for solving equations, inequalities, and many other algebra problems.