Problem 70 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(w^{2}+10 w+21\)

Step-by-Step Solution

Answer

The factorized form is \((w+3)(w+7)\).

Step 1 - Identify coefficients

In the trinomial, identify the coefficients. Here, the trinomial is \(w^{2}+10w+21\). The coefficients are: \(b = 10\) and \(c = 21\).

Step 2 - Find factor pairs of the constant term

List the factor pairs of the constant term, 21. These pairs are (1, 21) and (3, 7).

Step 3 - Determine the correct factor pair

Find the pair that adds up to the coefficient of the middle term, 10. The pair (3, 7) works since \(3 + 7 = 10\).

Step 4 - Write the factorized form

Using the factor pair (3, 7), write the factorized form as \((w+3)(w+7)\).

Key Concepts

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators like addition and subtraction. For example, the expression in our exercise is a trinomial: \( w^2 + 10w + 21 \). Understanding the components of an algebraic expression is important for solving problems involving factoring.
There are different types of algebraic expressions:

• Monomials (e.g., \(5x\))
• Binomials (e.g., \(x + 5\))
• Trinomials (e.g., \(w^2 + 10w + 21\))

In this exercise, we're dealing with a trinomial. The goal is to break it down into simpler binomials. Knowing how to identify the parts of these expressions makes it much easier to manipulate them in algebra.

trinomial factorization

Trinomial factorization is the process of breaking down a trinomial into two binomials. In our example, we start with the trinomial \(w^2 + 10w + 21\) and factor it into \((w+3)(w+7)\). Here’s how you do it:

1. Identify the coefficients as shown in Step 1.
2. Find factor pairs of the constant term, which are shown in Step 2.
3. Pick the factor pair that sums up to the middle coefficient, as in Step 3.
4. Write the factorized form according to Step 4.

Factorization simplifies the original expression and allows for easier solving of equations or further algebraic manipulation. It is crucial for solving quadratic equations and understanding polynomial behavior.

quadratic equations

Quadratic equations are polynomial equations of degree two, generally in the form \(ax^2 + bx + c = 0\). In dealing with the given exercise, we are working to understand and manipulate a specific quadratic equation: \(w^2 + 10w + 21\). Quadratic equations can be solved using several methods:

• Factoring (like in our exercise)
• Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Completing the square
• Graphing

Factoring is a straightforward method when the equation fits the form \(x^2 + bx + c\). Mastery of this basic form sets a solid foundation for tackling more complex quadratic equations.

polynomial factorization

Polynomial factorization extends beyond trinomials to include expressions of higher degrees. Understanding polynomial factorization lets you break down more complex expressions into products of simpler ones. Here are some common methods for polynomial factorization:

• Factoring out the greatest common factor (GCF)
• Grouping
• Special products (difference of squares, perfect square trinomials)
• Using factor theorems and synthetic division for higher-degree polynomials

Applying these techniques can help solve equations, simplify expressions, and understand polynomial functions. Mastering trinomial factorization is a key step in becoming proficient in polynomial factorization overall.