Problem 106 - Introductory Algebra for College Students

Question

Factor completely. (Hint on Exercises \(97-102\) : Factors contain rational numbers.) Divide \(x^{3}+4 x^{2}-3 x-18\) by \(x-2 .\) Use the quotient to factor \(x^{3}+4 x^{2}-3 x-18\) completely.

Step-by-Step Solution

Answer

The factors of \(x^{3}+4 x^{2}-3 x-18\) are \(x-2\) and \((x+3)^2\).

Step 1: Divide the Polynomials

First, use polynomial division to divide \(x^{3}+4 x^{2}-3 x-18\) by \(x-2\). For this polynomial division, place \(x^{3}+4 x^{2}-3 x-18\) under the division symbol and \(x-2\) to the left. Perform the division as you would with long division, keeping in mind the rules for adding, subtracting and multiplying polynomials.

Step 2: Calculate Quotient

Once you perform the division, the quotient you will obtain is \(x^2+6x+9\).

Step 3: Factor the Quotient

Now, factorize the quotient \(x^2+6x+9\) completely. This is a perfect square trinomial and can be written as \((x+3)^2\).

Step 4: State Factors of Given Polynomial

Since the given polynomial \(x^{3}+4 x^{2}-3 x-18\) is divided by \(x-2\) to get \(x^2+6x+9\), the factors of \(x^{3}+4 x^{2}-3 x-18\) are \(x-2\) and \((x+3)^2\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial, similar to long division with numbers. In the given exercise, we divide the polynomial \(x^3 + 4x^2 - 3x - 18\) by \(x - 2\) using this technique. The division process involves simplifying the polynomial step by step by:

• Aligning the terms according to their degree.
• Continuing through the division by writing the highest degree term first and carrying out the division for each term subsequently.
• Subtracting and bringing down the next terms as you would in number division.

The result of this division gives us a quotient polynomial, \(x^2 + 6x + 9\), which makes further factoring simpler. Understanding this division helps identify the factors and behavior of polynomials.

Perfect Square Trinomial

Once the polynomial \(x^3 + 4x^2 - 3x - 18\) is divided by \(x-2\) and the quotient \(x^2 + 6x + 9\) is found, we recognize it as a perfect square trinomial. This special type of polynomial has the form \(a^2 + 2ab + b^2\), and factors neatly into \((a + b)^2\).

• In \(x^2 + 6x + 9\), if we compare it with the form of a perfect square, we recognize that \(a\) is \(x\) and \(b\) is 3.
• This means that the polynomial can be factored as \((x+3)^2\).

Recognizing perfect square trinomials can greatly simplify the process of factoring, providing a quick and easy solution.

Rational Roots Theorem

The Rational Roots Theorem helps identify potential rational roots of a polynomial equation, which are crucial in simplifying and solving polynomials. Although not directly used in the steps for this exercise, it is worth mentioning as it provides insight into determining possible factors.

• According to the theorem, any rational root of the polynomial can be expressed as a fraction \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
• In our polynomial \(x^3 + 4x^2 - 3x - 18\), possible rational roots can be found by considering factors of \(-18\) (the constant) and 1 (the leading coefficient).

By testing these rational roots, we can verify potential factors before proceeding with polynomial division, thus confirming solutions or simplifying polynomial expressions.