Problem 71 - Algebra 2

Question

Factor each polynomial. $$ x^{2}+3 x+2 $$

Step-by-Step Solution

Answer

The polynomial factors to \((x + 1)(x + 2)\).

Step 1: Identify the form of the quadratic

The given quadratic polynomial is in the standard form \(ax^2 + bx + c\) where \(a = 1\), \(b = 3\), and \(c = 2\).

Step 2: Find two numbers that multiply to c and add to b

We need to find two numbers that multiply to \(c = 2\) and add to \(b = 3\). Consider the factors of 2: 1 and 2. These numbers satisfy both conditions (1 * 2 = 2 and 1 + 2 = 3).

Step 3: Write the factors using the numbers found

Using the numbers from Step 2, the factors of the polynomial can be written as \((x + 1)(x + 2)\). This is based on the idea that these numbers provide the original polynomial when expanded.

Step 4: Verify by expanding the factors

Expand \((x + 1)(x + 2)\) to verify: \[(x + 1)(x + 2) = x(x + 2) + 1(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2\]. This confirms that the factorization is correct.

Key Concepts

Quadratic Polynomials

Quadratic polynomials are a special type of polynomial that have a degree of 2. This means the highest exponent of the variable, usually described as \(x\), is 2. Quadratic polynomials take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
This form is essential to identify because it informs how we approach the problem of factorization. Quadratics are prevalent in various applications, such as physics for motion equations and economics in cost curves.
Working with quadratic polynomials involves operations such as addition, subtraction, multiplication, and factorization, the latter being a method to express the polynomial as a product of factors, which simplifies solving equations.

Standard Form

Standard form is an important concept when dealing with quadratic polynomials. It highlights the organized structure of the expression \(ax^2 + bx + c\). Here, \(a\) is the coefficient of the squared term \(x^2\), \(b\) is the coefficient of the linear term \(x\), and \(c\) is the constant term.

• The "standard form" allows for easy recognition of the coefficients \(a\), \(b\), and \(c\), which are crucial in factorization.
• It enables the application of various methods for solving, such as factoring or using the quadratic formula.

For example, in \(x^2 + 3x + 2\), the coefficients are \(a = 1\), \(b = 3\), and \(c = 2\). Recognizing these coefficients is the first step in the factorization process.

Polynomial Factorization

Polynomial factorization refers to breaking down a polynomial into simpler polynomials, known as factors, that when multiplied together result in the original polynomial. This process is analogous to breaking down a number into its prime factors.

• The goal in factorizing quadratic polynomials is to express them as the product of two binomials. This helps in resolving equations and understanding the behavior of polynomials.
• In our example of \(x^2 + 3x + 2\), we look for two numbers that multiply to \(c\) (which is 2), and add to \(b\) (which is 3). Here, those numbers are 1 and 2, as they satisfy both conditions.

The factorized form, \((x + 1)(x + 2)\), when multiplied out will reconstruct the original polynomial, showing the correctness of the factorization.

Verifying Factorization

Verifying factorization is a critical step to ensure accuracy in expressing a polynomial as a product of factors. This involves multiplying the factors back to see if they reproduce the original polynomial expression.
In our example, the factorization \((x + 1)(x + 2)\) is expanded using the distributive property:
- First, expand \(x(x + 2) = x^2 + 2x\).- Next, expand \(1(x + 2) = x + 2\).- Then, combine all terms, resulting in \(x^2 + 3x + 2\).
Successful verification confirms that no mistakes were made in finding and writing the factors. Ensuring the factorization is correct not only builds confidence but also ensures that subsequent problem-solving steps are based on correct assumptions.