Problem 7 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(x^{2}+5 x+3\) cannot be factored because we cannot find two integers whose product ____ is and whose sum is ____.

Step-by-Step Solution

Answer

The product is 3, and the sum is 5.

Step 1: Understand the Quadratic Expression

The quadratic expression given is \( x^2 + 5x + 3 \). In this context, we are looking to factor it into two binomials of the form \((x + m)(x + n) = x^2 + 5x + 3\).

Step 2: Identify Coefficients

Identify the coefficients from the expression: \( a = 1 \), \( b = 5 \), and \( c = 3 \). Here, \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.

Step 3: Set the Criteria for Factoring

For \( x^2 + 5x + 3 = (x+m)(x+n) \), the integers \( m \) and \( n \) must satisfy two conditions: \( m+n=5 \) (sum of the coefficients of \( x \)) and \( m \cdot n=3 \) (product of the constant term).

Step 4: Check Possible Integer Values

List pairs of integers whose product is 3: \((1, 3), (-1, -3), (3, 1), (-3, -1)\). None of these pairs sum to 5. Therefore, there are no integer pairs \((m, n)\) that satisfy both conditions.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) cannot be zero. The expression is termed 'quadratic' because the highest power of the variable \( x \) is 2. Knowing how to work with quadratic equations is fundamental in algebra.
Key Characteristics:

• The term \( ax^2 \) is known as the quadratic term.
• \( bx \) is called the linear term.
• The constant term is \( c \).

When you solve a quadratic equation, you are essentially finding the values of \( x \) that make the entire equation equal to zero. These solutions are also known as the roots of the equation. There are various methods available for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Understanding how to factor quadratics, when possible, allows for quicker and more intuitive solutions.

Product and Sum Method

The Product and Sum Method is a straightforward approach to factoring quadratic expressions, specifically those that can be written as a product of two binomials. This method only works well when the quadratic can be factored with integer coefficients.
How It Works:

• Start with a quadratic expression \( ax^2 + bx + c \).
• For factoring, find two numbers \( m \) and \( n \) such that:
  • The product \( m \cdot n = a \cdot c \).
  • The sum \( m + n = b \).

If both conditions above are met, the quadratic can be expressed as \((x + m)(x + n)\). However, not every quadratic expression can be factored using integers. If integer solutions can't be found, other methods like completing the square or using the quadratic formula should be considered.

Integer Coefficients

Dealing with integer coefficients means you're working with numbers in the quadratic equation that are whole numbers. These coefficients form the backbone of simpler algebraic expressions and can affect how we approach solving or factoring the equation.
Factors to Consider:

• Integer coefficients are easier to manage when performing arithmetic operations such as addition, subtraction, or multiplication.
• Using integers makes finding factors straightforward as you simply look for whole number solutions.
• If integers can't satisfy the conditions needed for the Product and Sum Method, it indicates that further advanced methods need to be employed.

Sometimes, factoring over integers might not be feasible, but understanding them provides a solid foundation for tackling more complex quadratic expressions.