Problem 12 - Precalculus

Question

Express each polynomial in the form \(a_{n}\left(x-r_{1}\right)\left(x-r_{2}\right) \cdots\left(x-r_{n}\right)\). $$x^{3}-2 x^{2}-3 x$$

Step-by-Step Solution

Answer

The polynomial can be expressed as \(x(x - 3)(x + 1)\).

Step 1: Factor Out the Greatest Common Factor (GCF)

Begin by identifying and factoring out the greatest common factor from all the terms in the polynomial. Here, the GCF of the terms \(x^3 - 2x^2 - 3x\) is \(x\). Thus, the polynomial can be factored as: \[x(x^2 - 2x - 3)\].

Step 2: Factor the Quadratic Polynomial

Now, focus on factoring the quadratic polynomial \(x^2 - 2x - 3\). Look for two numbers whose product is the constant term \(-3\) and whose sum is the coefficient of the linear term \(-2\). These numbers are \(-3\) and \(1\). Thus, \(x^2 - 2x - 3\) factors into \((x - 3)(x + 1)\).

Step 3: Write the Complete Factored Form

Combine the factor from Step 1 (the GCF) with the factors from Step 2. The polynomial \(x(x^2 - 2x - 3)\) then becomes \(x(x - 3)(x + 1)\).

Step 4: Identify the Coefficient and Roots

The coefficient in front (\(a_n\)) is \(1\) since \(x\) is the GCF factored out. The roots \(r_1, r_2,\) and \(r_3\) are \(0\), \(3\), and \(-1\) respectively, derived from the factors \(x\), \((x - 3)\), and \((x + 1)\).

Key Concepts

Understanding the Greatest Common Factor

One of the fundamental concepts of polynomial factorization is the greatest common factor, or GCF. The GCF is the largest factor shared by all terms in a given expression. By identifying the GCF, we can simplify polynomials, making them easier to solve or further factor.

Here's how you can find the GCF:

• Examine each term separately; for the polynomial \(x^3 - 2x^2 - 3x\), these terms are \(x^3\), \(-2x^2\), and \(-3x\).
• Focus on the variable and the coefficients. Commonly, you're looking for the highest power of any variable that all terms share. In this polynomial, each term includes a factor of \(x\).
• Strip out the \(x\) from each term, leaving behind \(x(x^2 - 2x - 3)\).

By factoring out the GCF, you simplify the polynomial, making it far easier to work with in subsequent steps. Remember, simplifying expressions or equations is a key component in not only solving them but also in understanding the underlying structure of polynomials.

Decoding Quadratic Polynomials

Quadratic polynomials come in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Factoring these polynomials involves finding two binomials that, when multiplied, will reproduce the original expression.

In the polynomial \(x^2 - 2x - 3\), we are looking for two numbers that:

• Multiply to \(-3\) (the constant term).
• Add to \(-2\) (the linear coefficient).

These numbers are \(-3\) and \(1\) because:

• \(-3 \times 1 = -3\)
• \(-3 + 1 = -2\)

Thus, it factors into \((x - 3)(x + 1)\). Quadratics may seem daunting at first, but using methods like product-sum can simplify the process. Practicing this technique will allow you to factorize quadratic polynomials more intuitively over time.

Look for simple pattern recognitions, like matching the signs first and then identifying the coefficients and constants. Mastering this crucial step is integral to polynomial factorization.

Exploring the Roots of Polynomials

Once a polynomial is completely factored, identifying its roots can become a straightforward task. Roots, or zeros, of a polynomial are the values of \(x\) that make the polynomial equal to zero.

For our factored polynomial \(x(x - 3)(x + 1)\):

• The factor \(x\) gives a root at \(x = 0\).
• The factor \(x - 3\) gives a root at \(x = 3\).
• The factor \(x + 1\) gives a root at \(x = -1\).

Knowing the roots helps understand where the graph of a polynomial intersects the x-axis. This insight is essential, especially when sketching graphs or solving polynomial equations.

Each root corresponds to one intersection point on the x-axis when graphed, offering valuable visual insight into the polynomial’s behavior and properties.