Problem 25 - Precalculus : Building Concepts and Connections

Question

Find all the real zeros of the polynomial. $$P(x)=x^{4}-13 x^{2}-12 x$$

Step-by-Step Solution

Answer

The real zeros of the polynomial \(P(x) = x^4 - 13x^2 - 12x\) are 0, 1, -4, and 3.

Step 1: Factor Out the Common Factor

Notice that all terms of the polynomial have \(x\) in common. Therefore, factor out an \(x\) from all the terms so that \$P(x) = x(x^3 - 13x - 12)$.

Step 2: Apply Factor Theorem

Next, apply the factor theorem to \(x^3 - 13x - 12\). The factor theorem states that a polynomial \(f(x)\) has a factor \(x - k\) if and only if \(f(k) = 0\). In this case, find factors \(a\), \(b\), and \(c\) such that \(a \cdot b \cdot c = -12\) and \(a + b + c = -13\). Notice that \(a=1\), \(b=-4\), and \(c=3\) meet the criteria. Therefore, the polynomial can be rewritten as \(x(x - 1)(x + 4)(x - 3)$.

Step 3: Apply Zero-Product Property

Finally, to find the real zeros of the polynomial, set each factor equal to zero: \(x = 0\), \(x - 1 = 0\), \(x + 4 = 0\), and \(x - 3 = 0\). Solving these equations gives \(x = 0\), \(x = 1\), \(x = -4\), and \(x = 3\), which are the real zeros of the polynomial.

Key Concepts

Factor Theorem

The Factor Theorem is an essential tool in algebra that connects the concept of polynomial factors to the roots or zeros of the polynomial. This theorem essentially tells us that if a number, say, \(k\), is a zero of a polynomial \(P(x)\), then \((x - k)\) is a factor of \(P(x)\). Conversely, if \((x - k)\) is a factor, then \(k\) is a root of the polynomial.

Applying the Factor Theorem is like solving a puzzle where you need to factor the polynomial completely to uncover all its roots. When you look at \(x^3 - 13x - 12\) in the original problem, using the theorem means finding values like \(x = 1\), \(x = -4\), and \(x = 3\) that make the polynomial equal zero. Thus, confirming each as a root means each corresponds to a factor \((x - 1)\), \((x + 4)\), and \((x - 3)\).

Using the Factor Theorem involves two key tasks:

• Identify a zero of the polynomial (either by trial and error or synthetic division).
• Rewrite the polynomial using the factor associated with this zero.

By doing so, you gain the ability to express a polynomial as a product of its factors, making it easier to solve for zeros.

Zero-Product Property

Once a polynomial is factored completely into the form of \( (x - k_1)(x - k_2)...(x - k_n) = 0 \), the Zero-Product Property becomes very helpful. This property states that if a product of factors equals zero, then at least one of the factors must be zero. Thus, each factor can be set equal to zero to find its roots.

In our polynomial \(x(x - 1)(x + 4)(x - 3)\), applying the Zero-Product Property means we set each factor \(x\), \((x - 1)\), \((x + 4)\), and \((x - 3)\) equal to zero. Solving these simple equations:

• \(x = 0\)
• \(x - 1 = 0\) gives \(x = 1\)
• \(x + 4 = 0\) gives \(x = -4\)
• \(x - 3 = 0\) gives \(x = 3\)

This approach not only helps to find the zeros of the polynomial but also allows us to verify that the polynomial has been factored correctly. By confirming that each factor corresponds to a root, we ensure our polynomial is effectively zeroed out at these points.

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into simpler polynomials, whose product is the original polynomial. This technique is crucial for finding the zeros of the polynomial, simplifying expressions, and solving polynomial equations.

To factor a polynomial effectively, follow these steps:

• Identify any common factors in all the terms of the polynomial. For example, in \(x^4 - 13x^2 - 12x\), \(x\) is a common factor, so we factor it out first: \(P(x) = x(x^3 - 13x - 12)\).
• Next, factor the remaining polynomial. This might involve recognizing patterns such as the difference of squares, perfect square trinomials, or using the Factor Theorem to identify zeros and corresponding factors.
• Continue factoring until no further factorization is possible, and check by multiplying the factors to ensure they give the original polynomial.

By mastering polynomial factoring, you not only make complex polynomials easier to manage but also lay the groundwork for solving equations effectively by revealing their roots.