Problem 45 - Elementary Algebra

Question

Factor. $$ x 2-16 x y+15 y 2 $$

Step-by-Step Solution

Answer

The factors are \((x - 15y)(x - y)\).

Step 1: Identify the Quadratic

The expression given is \( x^2 - 16xy + 15y^2 \). This is a quadratic expression in terms of \( x \) with \( y \) being a constant coefficient.

Step 2: Recognize the Standard Form

The standard form of a quadratic equation is \( ax^2 + bx + c \). Compare it to the given expression \( x^2 - 16xy + 15y^2 \) where \( a = 1 \), \( b = -16y \), and \( c = 15y^2 \).

Step 3: Set Up For Factoring

To factor the quadratic, look for two numbers whose product is \( ac = 15y^2 \) and the sum is \( b = -16y \). These numbers will help break down the expression for factoring.

Step 4: Find Suitable Numbers

The numbers that multiply to \( 15y^2 \) and add to \( -16y \) are \( -15y \) and \( -y \). This works because \( (-15y) imes (-y) = 15y^2 \) and \( (-15y) + (-y) = -16y \).

Step 5: Rewrite the Middle Term

Rewrite the expression by breaking the middle term using the numbers found: \( x^2 - 15xy - xy + 15y^2 \).

Step 6: Group the Terms

Group the terms: \((x^2 - 15xy) + (-xy + 15y^2)\).

Step 7: Factor by Grouping

Factor each group separately: \[ x(x - 15y) - y(x - 15y) \].

Step 8: Combine the Factors

Notice the common factor \((x - 15y)\) in both groups: \((x - 15y)(x - y)\). This is the factored form of the expression.

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial that includes a term with a variable raised to the power of two. In simpler terms, it's an expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. The highest power of the variable, usually \( x \), is 2, making it a quadratic.

• In the given exercise, the expression is \( x^2 - 16xy + 15y^2 \).
• This is a quadratic expression in terms of \( x \), where \( y \) acts as a constant coefficient.

Understanding that the variable \( x \) is squared helps us recognize the expression as quadratic, a fundamental step in the factoring process.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four terms. The key is to rearrange the polynomial into pairs of terms and then factor out the greatest common factor from each pair.

• Start by grouping the terms of the polynomial: - For \( x^2 - 16xy + 15y^2 \), we reorganize it as \((x^2 - 15xy) + (-xy + 15y^2)\).
• Each pair is then factored separately:
- In the first group \( (x^2 - 15xy) \), \( x \) is the common factor. So, it becomes \( x(x - 15y) \). - In the second group \( (-xy + 15y^2) \), \( -y \) is the common factor. So, it becomes \( -y(x - 15y) \).

The crucial step is identifying a common factor among both group results, which in this case is \( (x - 15y) \). This then allows for the completely factored expression to be formed: \((x - 15y)(x - y)\).

Polynomial

A polynomial is a mathematical expression made up of variables and constants combined using addition, subtraction, multiplication, and exponents that are whole numbers.

• In our example, \( x^2 - 16xy + 15y^2 \) is a polynomial because it involves variables \( x \) and \( y \) raised to a whole number power.
• Polynomials can have one or many terms. The term count influences which factoring method might be most efficient.
• For this exercise, note that a polynomial's complexity reduces when effectively factored, making it more manageable.

Recognizing the polynomial's structure and terms assists in choosing the appropriate method for solving or simplifying the expression.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \( ax^2 + bx + c \). Each part of this form has a specific role in the identity and behavior of the equation.

• \( a \) is the leading coefficient, determining the parabola's direction when graphed.
• \( b \) influences the symmetry and position of the parabola.
• \( c \) is the constant term, often the y-intercept on a graph.

For expression \( x^2 - 16xy + 15y^2 \), we identify \( a = 1 \), \( b = -16y \), and \( c = 15y^2 \). Knowing this form simplifies understanding the quadratic's features and guides the factoring approach. Comparing polynomials to this standard form makes them easier to interpret and manipulate mathematically.