Problem 19 - Beginning and Intermediate Algebra

Question

Factor by grouping. 2 r^{2}+11 r+15

Step-by-Step Solution

Answer

The factored form of the given expression \(2r^{2}+11r+15\) by grouping is \((2r+5)(r+3) \).

Step 1: Find two numbers that multiply to 30 and add to 11

We need to find two numbers that multiply to 2*15 = 30 and add to 11. The two numbers are 5 and 6.

Step 2: Rewrite the middle term

Rewrite the middle term using the two numbers (5 and 6) found in Step 1: \(2r^2 + 5r + 6r + 15\)

Step 3: Group terms

Group the terms in pairs, looking for a common factor in each pair: \((2r^2 + 5r) + (6r + 15)\)

Step 4: Factor out the common factor

Factor out the common factor from each group: \((r(2r + 5)) + (3(2r + 5))\)

Step 5: Factor by grouping

Notice that \((2r + 5)\) is a common factor in both terms. Factor it out: \((2r + 5)(r + 3)\) So, the factored form of the given expression by grouping is \((2r + 5)(r + 3)\).

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. Think of it like splitting a big number into its prime factors. This process helps in solving equations, simplifying expressions, and finding roots.

When you have a polynomial like \(2r^2 + 11r + 15\), you can often make it simpler by grouping terms and finding common factors. In this case, factored forms such as \((2r + 5)(r + 3)\) make the expression easier to work with.

The aim of factorization is to transform a more complex expression into a product of lower-degree polynomials, which are much easier to manage. By doing this, you create a simpler path for solving equations or simplifying expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition or multiplication) that represent a particular quantity. In our example, \(2r^2 + 11r + 15\), we have a blend of variables and coefficients. These expressions can be simplified using various techniques, one of which is factoring.

Here's what you need to remember about algebraic expressions:

• They are made of terms. Each term consists of a coefficient and a variable raised to a power.
• They can represent real-world situations, like calculating areas or solving problems.
• Simplifying them can make solving related equations much easier.

By transforming an algebraic expression into a factorized form, you can reveal its roots or solutions, which is invaluable for solving equations.

Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). These types of equations are fundamental in algebra and appear in numerous practical scenarios, like calculating projectile motion or optimizing areas.

In the example \(2r^2 + 11r + 15\), we are dealing with a quadratic expression. Factoring this into \((2r + 5)(r + 3)\) helps us quickly find its roots—values for \(r\) that make the equation equal zero.

To solve quadratic equations by factoring:

• Write the equation in its standard form.
• Use factorization techniques to break it into simpler binomials.
• Set each binomial to zero and solve for the variable.

Understanding how to factor quadratics helps not only in solving equations but also in graphing parabolas and analyzing their properties.