Problem 61 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$9 x^{4}+18 x^{3}+6 x^{2}$$

Step-by-Step Solution

Answer

The polynomial \(9x^4+18x^3+6x^2\) is completely factorized as \(3x^2 (3x^2 + 6x + 2)\)

Step 1: Identify and Pull out the common factor

First, identify the greatest common factor in all terms. In this case, the greatest common factor is \(3x^2\). Pull out this factor by dividing all terms by this factor. The result would be: \(3x^2 (3x^2 + 6x + 2)\)

Step 2: Check if factored polynomial can be factored further

Next, the term within the parenthesis, \(3x^2 + 6x + 2\), should be checked to see if it can be factored further. This polynomial happens to be a quadratic trinomial that cannot be factored with integer coefficients. Hence, it is prime.

Step 3: Present the final factorized form

Having verified that we can't factorize the trinomial further, we can thereby state that the complete factorization of our polynomial is \(3x^2 (3x^2 + 6x + 2)\).

Key Concepts

Greatest Common Factor

In the world of polynomials, identifying the Greatest Common Factor (GCF) is often the initial step when tackling a factoring problem. Think of the GCF as the largest expression that can be evenly divided into all terms of a polynomial. In our problem, we start with the expression \(9x^4 + 18x^3 + 6x^2\).
To find the GCF:

• First, examine the coefficients: 9, 18, and 6. The largest number that can divide all of them is 3.
• Next, for the variables, look at the lowest power of \(x\), which is \(x^2\) since the smallest exponent is 2.

Combining these, the GCF is \(3x^2\). To factor the polynomial, divide each term by \(3x^2\), pulling it out as a common factor. Applying this, you get \(3x^2 (3x^2 + 6x + 2)\). This step simplifies and reduces the terms, making further analysis easier.

Quadratic Trinomial

A quadratic trinomial is an expression consisting of three terms, typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratic trinomials are found frequently in algebra problems. In the polynomial \(3x^2(3x^2 + 6x + 2)\), the expression inside the parenthesis, \(3x^2 + 6x + 2\), represents such a trinomial.
To determine if this quadratic trinomial can be factored further, check if it can be expressed as a product of two binomials. This involves:

• Finding two numbers that multiply to the product of \(a\) and \(c\) (in our case, \(3 \times 2 = 6\)).
• These numbers also need to add up to \(b\), which is 6 in this expression.

Unfortunately, for \(3x^2 + 6x + 2\), there are no such integers that satisfy both conditions, meaning this trinomial cannot be factored further using integers.

Prime Polynomial

A prime polynomial is similar to a prime number in that it cannot be factored into simpler polynomials with integer coefficients. After identifying the GCF and extracting it from \(9x^4 + 18x^3 + 6x^2\), we are left with \(3x^2 + 6x + 2\). This is where the test for a prime polynomial comes into play.
Checking if a polynomial is prime involves:

• Attempting to factor the polynomial as shown in the quadratic trinomial section.
• If no factorization is possible with integer coefficients, it verifies the polynomial's primality.

In our exercise, the quadratic trinomial \(3x^2 + 6x + 2\) is considered prime. There are no two binomials with integer coefficients that multiply to this trinomial, confirming it as a prime polynomial. This means our final expression, \(3x^2(3x^2 + 6x + 2)\), is fully factored.