Problem 55 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(4 x^{3}-9 x^{2}-9 x\)

Step-by-Step Solution

Answer

The trinomial factors are \(x(4x + 3)(x - 3)\).

Step 1: Identify common factors

Look for a common factor in all terms of the trinomial. Here, each term contains an \(x\) as a factor. So, we can factor \(x\) out:\[4x^3 - 9x^2 - 9x = x(4x^2 - 9x - 9)\]

Step 2: Factor the quadratic trinomial

Now, focus on factorizing the quadratic expression \(4x^2 - 9x - 9\). We need to find two numbers that multiply to \(4 \times -9 = -36\) and add up to \(-9\). These numbers are \(-12\) and \(3\). Use these numbers to split the middle term:\[4x^2 - 12x + 3x - 9\]

Step 3: Group terms

Regroup the quadratic expression by pairing terms to factor by grouping:\[ (4x^2 - 12x) + (3x - 9) \]

Step 4: Factor by grouping

Factor out the greatest common factor in each pair of terms: \[4x(x - 3) + 3(x - 3)\]Observe that \(x - 3\) is a common factor.

Step 5: Complete Factorization

Factor out the common binomial factor \(x - 3\):\[(4x + 3)(x - 3)\]Now the trinomial is completely factored as:\[x(4x + 3)(x - 3)\]

Step 6: Verify the factorization

Ensure the factorization is correct by expanding the factored expression:Begin with \((4x + 3)(x - 3)\):\[4x^2 - 12x + 3x - 9 = 4x^2 - 9x - 9\]Then multiply by the \(x\) factored out initially:\[x(4x^2 - 9x - 9)\]It confirms that the original expression is obtained. Thus, the factorization is accurate.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this exercise, algebra helps us express mathematical relationships through equations and formulas. When factoring trinomials, algebraic skills are essential as you need to apply various techniques to simplify and manipulate expressions.

Algebra allows you to transform complex expressions into simpler forms and solve equations by isolating variables.
In the trinomial provided, operations like combining like terms, distributing, and factoring are guided by algebraic principles. Keep in mind that algebra is not just about solving for variables; it's about understanding how mathematical relationships work, providing a toolkit to tackle a wide range of problems. When breaking down expressions such as \(4x^3 - 9x^2 - 9x\) into more manageable parts, algebra looks at both structure and operations. This understanding is fundamental when moving on to more complex topics like polynomial equations.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials that are multiplied together. It's a crucial step in solving polynomial equations as it simplifies these equations, making them easier to manage and solve.
In our exercise, we begin by identifying a common factor among the terms. Recognizing that each term shares an \(x\), we factor this out first. Factorization doesn't stop there; after extracting common factors, you often need to dig deeper by focusing on quadratic expressions, another type of polynomial.

• Extract common factors like an \(x\) if it's present in all terms.
• Use various methods such as grouping or the quadratic formula for further breakdown.

Polynomial factorization can turn daunting expressions into a series of simpler, multiplied terms. This is not only useful for simplifying expressions but also for solving polynomial equations and finding roots.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically in the form \(ax^2 + bx + c\). They are standard components in many algebra problems, including our exercise, where we factor the quadratic \(4x^2 - 9x - 9\).
When factoring such expressions, the goal is to express them as a product of two binomials \((px + q)(rx + s)\). The challenge is to find numbers \(p, q, r,\) and \(s\) that satisfy the conditions resulting from expanding these binomials back into the quadratic form.

• Identify numbers that multiply to \(a\times c\) and add to \(b\).
• Split the middle term \(bx\) based on these numbers for further factorization.

Mastering quadratic expressions involves practicing these techniques until you can efficiently recognize patterns and factor quickly. This skill is pivotal in higher algebra and calculus.

Factoring by Grouping

Factoring by grouping is a strategic approach in algebra where terms in an expression are rearranged and grouped to make factorization straightforward. In complex trinomials, it often makes the process manageable where simple factorization isn't immediately obvious.

The process begins by pairing terms into groups that hold a common factor. In our exercise, we split the quadratic \(4x^2 - 12x + 3x - 9\) into two groups: \((4x^2 - 12x)\) and \((3x - 9)\). The idea is to factor out common factors within these groups:

• Group terms to allow extraction of the greatest common factors.
• Reorganize the expression successively until a common binomial emerges.

These steps bring out a common binomial, such as \(x - 3\), which can then be factored further. Factoring by grouping requires attention to detail and practice in recognizing viable groupings that facilitate further simplification.