Problem 38 - Algebra for College Students

Question

Factor completely. $$9 x^{2}-17 x^{4}+21 x^{5}$$

Step-by-Step Solution

Answer

The expression factors to \(x^2 (9 - 17x^2 + 21x^3)\); the trinomial isn't further factorable.

Step 1: Identify the Greatest Common Factor (GCF)

First, we identify the greatest common factor (GCF) of all the terms. In the expression \(9x^2 - 17x^4 + 21x^5\), each term has an \(x^2\). Therefore, the GCF is \(x^2\).

Step 2: Factor Out the GCF

Next, factor \(x^2\) out of each term:\[x^2(9 - 17x^2 + 21x^3)\]

Step 3: Factor the Trinomial

Now we focus on factoring the trinomial \(9 - 17x^2 + 21x^3\). Reorder it to \(21x^3 - 17x^2 + 9\) to have a more conventional polynomial order.

Step 4: Check for Factorable Trinomials

After reordering, we use trial and error, or the decomposition method if possible, to factor the polynomial. We look for two numbers that multiply to \(21 \times 9 = 189\) and add to \(-17\). Unfortunately, the trinomial is not directly factorable using integers.

Step 5: Combine Factors (if applicable)

Since the trinomial does not further factor over integers, we are left with the expression:\[x^2 (9 - 17x^2 + 21x^3)\] as the non-factorable form.

Key Concepts

Greatest Common Factor (GCF)

Finding the greatest common factor (GCF) is the first and most essential technique in the factorization of polynomial expressions. It involves identifying the biggest expression that divides each term in the polynomial equally without leaving a remainder. For the expression \(9x^2 - 17x^4 + 21x^5\), the focus is on the powers of \(x\). Each term in this polynomial shares \(x^2\) as a common factor. By factoring \(x^2\) out, we simplify the polynomial to a manageable form.

• Step 1: Write down all terms: \(9x^2\), \(-17x^4\), \(21x^5\).
• Step 2: Identify the smallest power of \(x\): \(x^2\).
• Step 3: The GCF is \(x^2\).

This simplification makes the polynomial easier to handle, setting a base for possible further factorization. It's like peeling a fruit to get to the juicy and interesting parts that might need more detailed exploration.

Trinomial Factorization

Trinomial factorization is an essential method for breaking down polynomials into simpler binomial factors. This process relies heavily on understanding the structure of the polynomial and using specific techniques such as trial and error, the product-sum method, or algebraic manipulation.In the given expression, after factoring out the GCF, the task becomes about simplifying \(21x^3 - 17x^2 + 9\). The rearranging of terms places it in descending power order, which is typical for handling polynomials. The absence of a straightforward integer factorization here highlights that not all trinomials are factorizable with integers. However, knowing how to search for pairs of numbers that multiply and sum aptly remains an invaluable strategy. Sometimes, numerical methods or specific factorization techniques could reveal non-obvious factors or ascertain irreducibility.

Polynomial Expressions

Polynomial expressions, such as \(9x^2 - 17x^4 + 21x^5\), are mathematical terms consisting of variables and coefficients arranged in a particular structure. Each polynomial is characterized by its degree (the highest power of the variable) and the number of terms it contains.Understanding the nature of polynomial expressions is vital, as they form the backbone of algebra. These expressions model real-world situations and have applications in various fields of science and engineering.

• Degree of a Polynomial: The highest power of the variable in the polynomial. For instance, \(9x^2 - 17x^4 + 21x^5\) has a degree of 5.
• Terms of a Polynomial: Each component separated by a plus or minus sign. In our example, there are three terms.

Polynomials can appear intimidating at first glance. However, with strategies like identifying the GCF and attempting to factor, they can be unraveled and solved methodically. Understanding their core components helps in efficiently manipulating these expressions to achieve clear, coherent solutions.