Problem 12 - Elementary and Intermediate Algebra

Question

Fill in each blank to explain how to factor \(x^{2}+7 x+10\) by grouping. We express the middle term, \(7 x,\) as the sum of _____ terms: $$x^{2}+7 x+10=x^{2}+\square x+\square x+10$$

Step-by-Step Solution

Answer

Use 2x and 5x to express the middle term.

Step 1: Identify the Product and Sum

To factor by grouping, first identify the need to express the middle term \(7x\) as the sum of two terms. Look for two numbers that multiply to the constant term \(10\) and add up to \(7\) (the coefficient of the middle term).

Step 2: Find the Pair

Listing factor pairs of \(10\) gives us \((1, 10)\) and \((2, 5)\). We select \(2\) and \(5\) because they add up to \(7\).

Step 3: Express Middle Term as Two Terms

Substitute the middle term \(7x\) by writing it as the sum \(2x + 5x\): \[x^2 + 7x + 10 = x^2 + 2x + 5x + 10\]

Key Concepts

Grouping Method

The grouping method is an effective strategy to factor quadratic expressions. It revolves around separating the expression into groups, allowing the factorization process to become more manageable and intuitive. When dealing with quadratic expressions in the form of \( ax^2 + bx + c \), the technique focuses on rewriting the middle term.To start, we need to divide the quadratic expression into two groups. Consider the quadratic equation \( x^2 + 7x + 10 \). Our goal is to express the middle term, \( 7x \), as a sum of two terms that make the expression easier to handle. This step prepares the expression for factorization through grouping, creating a smooth path toward simplifying the polynomial.

Factoring by Grouping

Factoring by grouping is a method that involves breaking down a polynomial into smaller, more manageable groups. These groups, once identified, can be factored individually, ultimately simplifying the entire expression. To illustrate, take the expression \( x^2 + 7x + 10 \). We express the middle term \( 7x \) as two separate terms, such as \( 2x \) and \( 5x \). This gives us:\[ x^2 + 2x + 5x + 10 \]Notice that we chose \( 2x \) and \( 5x \) because the product of the terms, when combined with \( x^2 \) and constant \( 10 \), creates a scenario where each group of terms can be factored by a common factor. The expression is now ready for the next step of factorization where terms in small groups can be factored out of the polynomial.

Polynomial Factorization

Polynomial factorization refers to the process of breaking down a polynomial into a product of its factors. It transforms the expression into a multiplication of simpler terms. This is particularly helpful when solving polynomial equations, making calculations more efficient.In the context of \( x^2 + 7x + 10 \), the factorization aims to convert it into a product of binomials. After expressing the middle term and grouping, the equation becomes:\[ (x^2 + 2x) + (5x + 10) \]From here, each group is factored separately:

• \( x(x + 2) \) from the first group \( x^2 + 2x \)
• \( 5(x + 2) \) from the second group \( 5x + 10 \)

The equation now reads:\[ x(x + 2) + 5(x + 2) \]The common factor \( x + 2 \) is further factored out:\[ (x + 2)(x + 5) \]Thus, the polynomial \( x^2 + 7x + 10 \) is factored into \( (x + 2)(x + 5) \), showing the power and simplicity of polynomial factorization.