Problem 65 - Algebra 2

Question

Which expression is the factored form of \(x^{3}+2 x^{2}-5 x-6\) ? $$\begin{array}{ll}{\text { A. }(x+1)(x+1)(x-6)} & {\text { B. }(x+2)(2 x-5)(x-6)} \\ {\text { C. }(x+3)(x+1)(x-2)} & {\text { D. }(x-3)(x-1)(x+2)}\end{array}$$

Step-by-Step Solution

Answer

The factored form of the polynomial is option C. \(x+3)(x+1)(x-2)\).

Step 1: Factor by Grouping

To begin factoring the polynomial, look for common factors or use techniques such as grouping. In this case, start by attempting to group the terms into pairs that can be factored separately.

Step 2: Find Possible Rational Roots

Use the Rational Root Theorem to list all possible rational roots of the polynomial, which are of the form \(\pm\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.

Step 3: Test the Possible Roots

Substitute the possible roots into the polynomial to identify which ones are actual roots. Once a root is found, it implies a linear factor of the polynomial.

Step 4: Factor Out the Linear Factor

When an actual root is identified, divide the polynomial by the corresponding linear factor to reduce the polynomial's degree.

Step 5: Factor the Quadratic

With the polynomial now being quadratic, factor it further, if possible, to find the remaining factors.

Step 6: Identify the Correct Factored Form

Now match the factors with the given multiple choice answers to select the correct factored form of the polynomial.

Key Concepts

Rational Root Theorem

Understanding the Rational Root Theorem is crucial when it comes to factoring polynomials and locating their roots. Simply put, this theorem provides us with a method to find all possible rational roots of a polynomial equation. This is particularly useful for higher degree polynomials where trial-and-error would be impractical.

The theorem states if a polynomial equation has rational roots, they are likely to be found among the ratios formed by the factors of the constant term and the factors of the leading coefficient. Mathematically, this is presented as \(\pm\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. For instance, if our polynomial has a constant term of 6 and a leading coefficient of 1, as in the provided exercise \(x^3 + 2x^2 - 5x - 6\), the possible rational roots would be \(\pm1\), \(\pm2\), \(\pm3\), and \(\pm6\).

Once these possible roots are listed, you can systematically test each one - typically starting with the smallest absolute value - in the original polynomial. If a value yields zero, it confirms that the number is indeed a root of the polynomial.

Factor by Grouping

Factoring by grouping is a technique applied to polynomials that allows us to break down the expression into smaller, more manageable parts. It's often used when a polynomial does not have an immediately obvious factoring pattern.

The process begins by rearranging and grouping the terms in such a way that each group has a common factor. In our expression \(x^3 + 2x^2 - 5x - 6\), this might involve regrouping the terms as \(x^2(x + 2) - 3(x + 2)\). Then, you would factor out the common binomial, which results in \( (x+2) \cdot (x^{2}-3) \).

It's essential to move step by step, ensuring that each grouping genuinely has a common factor. If done properly, this technique can greatly simplify the search for polynomial roots, especially when combined with other methods like the Rational Root Theorem.

Polynomial Roots

Polynomial roots, often called zeros or x-intercepts in a graphical context, are the values for which the polynomial is equal to zero. For a given polynomial equation, \( P(x) = 0 \), the roots are the solutions to this equation. They hold the key to factoring the polynomial into its component linear factors.

Knowing the roots allows us to write the polynomial as the product of its factors. For example, if the roots of \(x^3 + 2x^2 - 5x - 6\) are \( r_{1}\), \( r_{2}\), and \( r_{3}\), the factored form of the polynomial would be \( (x - r_{1})(x - r_{2})(x - r_{3}) \). These roots could be either real or complex numbers. In our exercise, the roots are real and can be determined systematically by using the Rational Root Theorem to list possible solutions and factor by grouping to simplify the polynomial into a product of factors, which leads to the identification of its roots.

Thoroughly understanding these concepts not only helps in solving polynomial equations but also in appreciating the broader implications of roots in calculus, where they play a significant role in sketching the curves of polynomial functions.