Problem 19 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$ P(x)=x^{3}+3 x^{2}+6 x+4 $$

Step-by-Step Solution

Answer

The only rational zero is -1.

Step 1: Identify Possible Rational Zeros

According to the Rational Root Theorem, any rational zero of a polynomial is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. For \( P(x) = x^3 + 3x^2 + 6x + 4 \), the constant term is 4 and the leading coefficient is 1. Thus, the possible rational zeros are \( \pm 1, \pm 2, \pm 4 \).

Step 2: Test Possible Zeros

To find the actual rational zeros, substitute each possible zero into \( P(x) \) to see if it equals zero. 1.\( P(1) = 1^3 + 3 \cdot 1^2 + 6 \cdot 1 + 4 = 1 + 3 + 6 + 4 = 14 \) (not a root) 2.\( P(-1) = (-1)^3 + 3(-1)^2 + 6(-1) + 4 = -1 + 3 - 6 + 4 = 0 \) (a root)3.\( P(2) = 2^3 + 3 \cdot 2^2 + 6 \cdot 2 + 4 = 8 + 12 + 12 + 4 = 36 \) (not a root) 4.\( P(-2) = (-2)^3 + 3(-2)^2 + 6(-2) + 4 = -8 + 12 - 12 + 4 = -4 \) (not a root)5.\( P(4) = 4^3 + 3 \cdot 4^2 + 6 \cdot 4 + 4 = 64 + 48 + 24 + 4 = 140 \) (not a root)6.\( P(-4) = (-4)^3 + 3(-4)^2 + 6(-4) + 4 = -64 + 48 - 24 + 4 = -36 \) (not a root)

Step 3: Verify and Conclude

From the calculations above, the only rational zero of \( P(x) \) is \(-1\). Therefore, the polynomial \( P(x) = x^3 + 3x^2 + 6x + 4 \) has one rational zero which is \(-1\).

Key Concepts

Polynomial Zeros

When tackling polynomial equations like \[P(x) = x^3 + 3x^2 + 6x + 4\], a key goal is to find the zeros or "roots" of the function. Zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. In simpler terms, these are the points where the graph of the polynomial crosses or touches the x-axis. Finding zeros is crucial because:

• They reveal important properties of the polynomial's graph.
• They help in factorizing the polynomial, making computations easier.
• Zeros are essential in solving polynomial equations and inequalities.

For rational zeros, we use the Rational Root Theorem, which gives a list of possible rational zeros that the polynomial might have. Testing these potential candidates helps us identify actual roots, leading us to factorize the polynomial further or to find its complete set of roots.

Factor Theorem

The Factor Theorem is a fundamental principle in algebra that links zeros of polynomials with their factors. According to this theorem, if \(x = a\) is a zero of the polynomial \(P(x)\), then \(x - a\) is a factor of \(P(x)\). In other words, having a polynomial zero means that the polynomial can be divided by a corresponding linear factor without leaving a remainder.This theorem allows:

• Simplifying the polynomial by breaking it down into smaller factors.
• Understanding the behavior of the polynomial by analyzing its factors.
• Identifying hidden patterns or symmetries in the polynomial function.

In our exercise, since \[P(-1) = 0\], we realize that \(x + 1\) is a factor of \(P(x)\). This factorization aspect not only simplifies polynomial functions but also provides a straightforward way to solve polynomial equations.

Synthetic Division

Synthetic division offers a fast and efficient method for dividing polynomials, especially useful when testing potential zeros found using the Rational Root Theorem. This method simplifies long division, specifically when the divisor is a linear factor \(x - a\).Using synthetic division:

• You can quickly determine whether a candidate zero is indeed a true zero.
• It provides the quotient and remainder, assisting in factorization of the polynomial.
• The process is less error-prone and quicker than traditional polynomial division.

To use synthetic division with our polynomial and \(x = -1\):1. Write down the coefficients of \(P(x)\): 1, 3, 6, 4.2. Conduct synthetic division with \(-1\).If the remainder is zero, \(-1\) is confirmed as a root, aligning with the Rational Root Theorem and Factor Theorem.It's a practical arithmetic shortcut that saves time and effort in polynomial calculations.