Problem 12 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 5 . \(36 r^{2}-5 r-24\)

Step-by-Step Solution

Answer

The trinomial factors into \((9r - 8)(4r + 3)\).

Step 1: Identify the trinomial form

The trinomial given is in the form \( ax^2 + bx + c \), where \( a = 36 \), \( b = -5 \), and \( c = -24 \).

Step 2: Multiply a and c

Multiply \( a \) and \( c \) to help in splitting the middle term. Calculate \( 36 \times (-24) = -864 \).

Step 3: Find two numbers that multiply to ac and add to b

We need two numbers that multiply to \(-864\) and add to \(-5\). The numbers are 27 and -32, since \( 27 \times (-32) = -864 \) and \( 27 + (-32) = -5 \).

Step 4: Rewrite the middle term using the two numbers

Rewrite \(-5r\) as \( 27r - 32r \). The trinomial becomes \(36r^2 + 27r - 32r - 24\).

Step 5: Group and factor by grouping

First, group terms: \((36r^2 + 27r) + (-32r - 24)\). Factor each group separately. From \(36r^2 + 27r\), factor out \(9r\) to get \(9r(4r + 3)\). From \(-32r - 24\), factor out \(-8\) to get \(-8(4r + 3)\).

Step 6: Factor out the common binomial factor

Both groups contain \(4r + 3\). Factor this out so that the expression becomes \((9r - 8)(4r + 3)\).

Step 7: Verify the factorization

Expand \((9r - 8)(4r + 3)\) to ensure it returns to \(36r^2 - 5r - 24\). The expansion gives \(36r^2 + 27r - 32r - 24\), simplifying to \(36r^2 - 5r - 24\).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. For example, in the expression \( 36r^2 - 5r - 24 \), the term \( 36r^2 \) is a quadratic term. Quadratic terms involve the variable raised to the power of two, signifying the presence of a squared element.
These expressions are the building blocks of algebra and help us define relationships mathematically.

• In algebraic expressions, each part separated by a plus or minus sign is called a term.
• The coefficient is the numerical part multiplying the variable in a term.
• The expression given above is specifically a trinomial, as it consists of three terms.

Performing operations on these expressions, like addition or multiplication, can help us solve equations or factor them into simpler parts.

Polynomial Factorization

Polynomial factorization is a method used to simplify polynomials like \( 36r^2 - 5r - 24 \) into a product of simpler polynomial expressions. This process is crucial because it can make algebraic equations more manageable and easier to solve.
By factorizing polynomials, we break them down into their original multiplicative components, similar to finding the prime factors of a number.

• The vocabulary for factorization includes terms such as factors, which are expressions that can be multiplied to get the original polynomial.
• The process of finding these factors is known as 'decomposing' or 'splitting' the polynomial.

In many cases, especially with quadratic expressions, factorization involves breaking down the middle term or grouping terms strategically.

Quadratic Equations

Quadratic equations are a type of polynomial equation that takes the form \( ax^2 + bx + c = 0 \). The equation \( 36r^2 - 5r - 24 = 0 \) is quadratic because it contains a term with \( r^2 \). These equations are solved in various ways, including factorization, completing the square, and using the quadratic formula.
One common method is finding the roots through factorization, where the original quadratic equation is expressed as a product of factors, often two binomials.

• Roots of quadratic equations are the solutions where the equation equals zero.
• The solutions can be real or complex numbers, and finding them helps to understand the behavior of the function.

Factoring is a powerful tool in algebra because it simplifies the path to solving these equations, especially when their roots are rational numbers.

Factoring Techniques

Factoring techniques are strategies used to rewrite a polynomial as a product of its factors. In the case of \( 36r^2 - 5r - 24 \), various steps are employed to successfully factor the trinomial.
Different factoring techniques include:

• Grouping: This involves rearranging terms and factoring in pairs, as demonstrated in the original solution. By grouping terms that share a common factor, you can simplify the polynomial.
• Splitting the middle term: This technique requires breaking down the middle term into two parts that allow for easier factoring through grouping.
• Finding common factors: Identify and factor out any greatest common factor among the terms.

Mastering these techniques ensures that you can efficiently factor a wide variety of polynomials, paving the way for solving more complex algebraic problems.