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Magazine Chapter 6: Problem 45
Factor. $$ x 2-16 x y+15 y 2 $$
Short Answer
Expert verified
The factors are \((x - 15y)(x - y)\).
Step by step solution
01
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.
02
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 \( c = 15y^2 \).
03
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.
04
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 \( (-15y) + (-y) = -16y \).
05
Rewrite the Middle Term
Rewrite the expression by breaking the middle term using the numbers found: \( x^2 - 15xy - xy + 15y^2 \).
06
Group the Terms
Group the terms: \((x^2 - 15xy) + (-xy + 15y^2)\).
07
Factor by Grouping
Factor each group separately: \[ x(x - 15y) - y(x - 15y) \].
08
Combine the Factors
Notice the common factor \((x - 15y)\) in both groups: \((x - 15y)(x - y)\). This is the factored form of the expression.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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 \( -y(x - 15y) \).
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.
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.
