Problem 71 - College Algebra

Question

Factor completely, or state that the polynomial is prime. $$ x^{3}+2 x^{2}-9 x-18 $$

Step-by-Step Solution

Answer

The factorized form of the given polynomial \(x^{3}+2 x^{2}-9 x-18\) is \((x+2)(x-3)^{2}\).

Step 1: Identify Possible Rational Roots

The possible rational roots of a polynomial equation can be determined by p/q, where p is a factor of the constant term (in this case -18), and q is a factor of the leading coefficient (in this case 1). For this equation, the possible rational roots are ±1, ±2, ±3, ±6, ±9, ±18.

Step 2: Perform Synthetic Division

Synthetic division is performed with each of the possible rational roots identified in the previous step until you find a rational root (a remainder of zero). In this case, after performing synthetic division, it is found that -2 and 3 are roots of the polynomial.

Step 3: Factorize the Polynomial

After identifying the roots -2 and 3, the original polynomial \(x^{3}+2 x^{2}-9 x-18\) can be expressed as \((x+2)(x-3)^{2}\), as the double root 3 would appear twice in the factorized form of the polynomial.

Key Concepts

Understanding Rational Roots

Polynomials often have roots that are rational numbers. To find these, we rely on the Rational Root Theorem. This theorem suggests that if a polynomial has a rational solution, which can often be expressed as \( \frac{p}{q} \), then: - \( p \) is a factor of the constant term (which is\(-18\) in our exercise) - \( q \) is a factor of the leading coefficient (here it's\(1\)) So, in simpler terms, possible rational roots are obtained by dividing the factors of the constant term by the factors of the leading coefficient. For our polynomial, these possible roots are ±1, ±2, ±3, ±6, ±9, and ±18. By checking each of these possible rational roots using synthetic division, we can efficiently determine which ones are actual roots of the polynomial.

Using Synthetic Division

Synthetic division is a handy shortcut for dividing a polynomial by a binomial. It simplifies the process significantly and is less cumbersome than long division. You usually use synthetic division when you want to test the possible rational roots found earlier. To perform synthetic division: - Write down the coefficients of the polynomial. For \(x^{3} + 2x^{2} - 9x - 18\), this would be\(1, 2, -9, -18\).- Choose a candidate root from the list of possible rational roots, like \(-2\) or \(3\), and arrange it near the chosen coefficients.- Follow the process of bringing down the leading coefficient, multiplying, then adding, moving through all coefficients. If the result yields a remainder of zero, the tested value is indeed a root of the polynomial. In this exercise, both \(-2\) and \(3\) give a remainder of zero, confirming they are roots.

Polynomial Factorization

Polynomials can often be expressed as a product of simpler polynomials. After identifying roots using synthetic division, factorization becomes clearer and straightforward. For our polynomial \(x^{3} + 2x^{2} - 9x - 18\), once we've confirmed the roots \(-2\) and \(3\), we can begin the factorization process:- Since \(-2\) is a root, it can be expressed as \((x + 2)\).- Meanwhile, \(3\) not only provides a root but appears twice as a double root, resulting in \((x - 3)^{2}\).Thus, the original polynomial can be fully factorized into \((x + 2)(x - 3)^{2}\). This step-by-step approach illustrates how to reduce the polynomial into simpler components, revealing its structure and solutions.