Problem 67 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(a^{2}+9 a+20\)

Step-by-Step Solution

Answer

(a + 4)(a + 5)

Step 1: Identify the coefficients

Identify the coefficients in the trinomial. Here, the trinomial is given by: \(a^2 + 9a + 20\).The coefficients are:- Coefficient of \(a^2\) is 1 (since there is no numerical coefficient, it is assumed to be 1)- Coefficient of \(a\) is 9- Constant term is 20.

Step 2: Find two numbers that multiply to the constant term and add to the linear coefficient

Find two numbers that multiply to 20 and add up to 9. These numbers are 4 and 5 because:-\(4 \times 5 = 20\)-\(4 + 5 = 9\)

Step 3: Write the factorized form

Using the numbers found in Step 2, write the factorized form of the trinomial. The factorized form is:\((a + 4)(a + 5)\)

Key Concepts

quadratic equations

Quadratic equations are a foundational concept in algebra and involve any equation that can be written in the form: \[ax^2 + bx + c = 0\]. Here, 'a', 'b', and 'c' are constants with 'a' not equal to zero. Quadratic equations have numerous real-world applications, from physics to economics.
Key Features:

• Parabolic Graph: The graph of a quadratic equation is a parabola, which can either open upward or downward depending on the sign of 'a'.
• Roots or Solutions: Quadratic equations can have two, one, or no real solutions. These roots can be found using the quadratic formula:
  \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
• Factoring: One method to solve quadratic equations is by factoring, which involves expressing the quadratic equation as a product of two binomials.

Understanding quadratic equations is crucial for higher-level mathematics and various practical problems.

polynomial factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give the original polynomial.
In our exercise, the trinomial \(a^2 + 9a + 20\) was factored into \((a + 4)(a + 5)\).
Key Steps in Factoring Trinomials:

• Identify Coefficients: Determine the coefficients for the terms. Here, the coefficients are 1 (for \(a^2\)), 9 (for 'a'), and 20 (constant term).
• Find Pair of Numbers: Find two numbers whose product is the constant term and whose sum is the linear coefficient. For \(a^2 + 9a + 20\), the numbers are 4 and 5.
• Write Product of Binomials: Use these numbers to write out the factorized form: \((a + 4)(a + 5)\).

Tips for Factoring:

• Check Your Work: Always multiply the binomials back out to ensure you got the correct factorization.
• Practice: The more trinomials you practice factoring, the more intuitive the process will become.

Factorization simplifies solving polynomials and is a powerful tool in algebra.

algebraic expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Unlike equations, they do not have an equal sign. Examples include \(3x + 4\) and \(a^2 + 9a + 20\).
Components of Algebraic Expressions:

• Variables: Symbols that represent numbers. In our example, 'a' is a variable.
• Coefficients: Numbers that multiply the variables. Here, 9 is the coefficient of 'a'.
• Constants: Numbers without variables. In \(a^2 + 9a + 20\), 20 is the constant term.

Why Understand Algebraic Expressions:

• Foundation: They are the building blocks for equations and inequalities.
• Simplifies Complex Problems: Understanding how to manipulate and factor expressions helps simplify complex algebraic problems.
• Real-world Applications: Used in calculating interest rates, physics problems, and optimizing functions in economics and engineering.

Mastering algebraic expressions lays the groundwork for success in higher mathematics and practical problem-solving scenarios.