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Chapter 6: Problem 51

Factor each trinomial completely. $$ 4 k^{3}-4 k^{2}+9 k $$

Short Answer

Expert verified
k(4k^2 - 4k + 9)

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Look at all the terms in the given trinomial, which are - 4k^3, - 4k^2, - 9k. Identify the greatest common factor (GCF) of these terms. The GCF is the highest factor that divides all terms. Here, it is k.
02

Factor out the GCF from the trinomial

Factor out k from each term to get: k(4k^2 - 4k + 9)
03

Simplify if possible

Now, check if the quadratic expression inside the parentheses can be further factored. In this case, the quadratic expression 4k^2 - 4k + 9 does not factor further since it doesn't have roots that can be expressed in a simpler product of binomials.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Greatest Common Factor (GCF)
To factor trinomials, we first need to understand the **Greatest Common Factor (GCF)**. The GCF is the largest number (or algebraic term) that can divide each term in the expression without leaving a remainder. To identify the GCF, look at the coefficients and variables of each term. For example, in the trinomial:

$$ 4k^3 - 4k^2 + 9k $$ The terms are 4k鲁, -4k虏, and 9k. The GCF of these terms is k because k is the largest factor that appears in each term. Finding the GCF simplifies the factoring process by reducing the expression, making it easier to work with.
Exploring Quadratic Expressions
A **Quadratic Expression** is a polynomial that takes the form *ax虏 + bx + c*. In our problem, after factoring out the GCF, we are left with the quadratic expression:

$$ 4k虏 - 4k + 9 $$

To factor trinomials (quadratic expressions), we typically look for two binomials that multiply to give the original expression. This involves finding factors of a, b, and c that satisfy the quadratic's structure. However, not all quadratic expressions are factorable by simple methods. If a quadratic cannot be factored further, it is termed 'prime.' In our case, 4k虏 - 4k + 9 does not have real roots that can be expressed neatly as binomials, hence it is prime.
Simplifying Algebraic Expressions for Clarity
After factoring out the GCF and simplifying, we reach a stage where the expression is easier to handle. Simplifying algebraic expressions by pulling out common factors reduces the complexity of the problem. For the given trinomial, we simplified as so:

$$ 4k^3 - 4k^2 + 9k $$ transforms to:

$$ k(4k^2 - 4k + 9) $$

Simplification helps in recognizing patterns and factors that are not immediately visible. It ensures we can work with reduced forms, which are easier to analyse and understand.

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Most popular questions from this chapter

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