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Chapter 6: Problem 36

Factor completely. If the polynomial cannot be factored, write prime. $$ x^{2}-13 x+36 $$

Short Answer

Expert verified
The factored form is \( (x-4)(x-9) \).

Step by step solution

01

Identify the quadratic equation

The given polynomial is a quadratic equation of the form \(ax^2 + bx + c \). In this case, it is \(x^2 - 13x + 36\).
02

Find factors of the constant term

Identify two numbers that multiply to the constant term (36) and add up to the coefficient of the middle term (-13). These two numbers will be the factors of the quadratic equation.
03

Determine the right pair of factors

Examine pairs of factors of 36: 1 and 36, 2 and 18, 3 and 12, 4 and 9, 6 and 6. Notice that the pair (-4) and (-9) multiply to 36 and add to -13.
04

Write the factored form

Use the identified factors to write the quadratic in factored form: \( (x-4)(x-9) \). Therefore, the completely factored form of \(x^2 - 13x + 36\) is \( (x-4)(x-9) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equation
A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In the given exercise, the quadratic equation is \( x^2 - 13x + 36 \). The highest exponent is 2, which tells us this is a quadratic equation.
Quadratic equations often appear in physics, engineering, and various real-world problems. Understanding how to solve them is crucial.
  • The term \( ax^2 \) is called the quadratic term.
  • The term \( bx \) is the linear term.
  • The term \( c \) is the constant term.
To factor the quadratic equation, we need to find two numbers that satisfy specific conditions related to these terms. This involves understanding both the product of the constant term and the sum of the coefficient of the linear term.
Factoring Polynomials
Factoring polynomials involves breaking down a polynomial into simpler 'factor' polynomials that multiply to give the original polynomial. In the problem \( x^2 - 13x + 36 \), we aim to express it as \( (x - m)(x - n) \), where \( m \) and \( n \) are numbers to be found.
Steps to factor a polynomial like this:
  • Identify the quadratic equation format \( ax^2 + bx + c \).
  • Find two numbers that multiply to the constant term \( c \) and add up to the coefficient \( b \) of the linear term.
  • Write the polynomial in its factored form using the identified numbers.
For the given equation \( x^2 - 13x + 36 \), we need to find two numbers that multiply to 36 and add up to -13. After examining pairs, we find \( -4 \) and \( -9 \) work. Therefore, the quadratic equation factors to \( (x - 4)(x - 9) \).
Constants and Coefficients
In a quadratic equation \( ax^2 + bx + c \), \( a \), \( b \), and \( c \) are fundamental elements known as coefficients and constants.

Let's define each one:
  • Coefficients: These are the numerical factors of the terms in a polynomial. In \( x^2 - 13x + 36 \), the coefficient of \( x^2 \) is 1 (since it's written as 1\( x^2 \)), and the coefficient of \( x \) is -13.
  • Constant: This is the term without any variable attached to it. In the equation, 36 is the constant.
The value of these coefficients and constants directly affects how we factor the polynomial. To summarize:
  • The constant term provides the product for the factor pairing.
  • The coefficient of the middle term gives the sum required of those factors.
Recognizing these components is essential for efficiently factoring quadratic equations and solving problems related to them in various mathematical and real-world applications. With practice, identifying the necessary numbers and writing the factored form becomes easier.

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