Problem 11 - Algebra and Trigonometry

Question

\(5-12\) . Factor out the common factor. $$ y(y-6)+9(y-6) $$

Step-by-Step Solution

Answer

The expression factors to \((y-6)(y+9)\).

Step 1: Identify the Common Factor

Examine the given expression to find a common factor in both terms. The expression is \( y(y-6) + 9(y-6) \). Notice that \((y-6)\) is present in both terms.

Step 2: Factor Out the Common Factor

Since \((y-6)\) is a common factor, we factor it out of the expression. This means we will write \((y-6)\) once and divide each term by \((y-6)\). Thus, we have: \[ (y-6)(y + 9) \].

Step 3: Rewrite the Expression

Rewrite the expression after factoring out the common factor. The factored form of the expression is now: \[ (y-6)(y + 9) \].

Key Concepts

Common Factor

When working with algebraic expressions, one efficient way to simplify them is by identifying and factoring out the common factor. A common factor is a term that appears in each part of the expression. In the given expression, \( y(y-6) + 9(y-6) \), notice that \((y-6)\) is repeated in both terms. This is the key step in factoring. By recognizing common factors, you can make expressions simpler and reduce them to more manageable forms. This is especially handy in polynomial expressions, where terms might share variables or constants as common factors.

Factoring Expressions

Factoring expressions involves rewriting an expression as a product of its factors. It's like breaking down a complicated expression into simpler pieces. For the expression \( y(y-6) + 9(y-6) \), after identifying the common factor \((y-6)\), we factor it out to get the factored expression \((y-6)(y + 9)\). The process requires recognizing patterns and applying distributive properties in reverse. This skill is crucial in algebra for simplifying equations, solving equations, and even finding the roots of polynomial functions. Here, once the expression is factored, it is ready for further analysis or use.

Polynomials

Polynomials are algebraic expressions that include variables raised to whole number exponents and coefficients. They can be of various forms, like linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on. Each term in a polynomial is a product of a constant and a non-negative power of a variable.The original expression \( y(y-6) + 9(y-6) \) is a polynomial since it's made up of terms with variables and constants. Understanding how to manipulate these polynomials, including factoring them, is an essential skill in algebra. By factoring, we can reveal key properties of polynomials such as their roots and intercepts.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They are the language of algebra, used to formulate equations, models, and various mathematical expressions. In the expression \( y(y-6) + 9(y-6) \), you see a perfect example of an algebraic expression. Once you start working with expressions, understanding how to simplify them by recognizing common factors or terms becomes crucial. Simplifying expressions through factoring makes the math less daunting and helps in solving algebraic problems more efficiently. This technique is not just useful in solving equations but is also widely used in calculus, physics, engineering, and more advanced mathematical fields.