Problem 86 - Algebra and Trigonometry

Question

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ 3 x^{3}+8 x^{2}+5 x+2=0 ;[-3,3] \text { by }[-10,10] $$

Step-by-Step Solution

Answer

The only rational root is \( x = -1 \).

Step 1: State the Rational Zeros Theorem

The Rational Zeros Theorem states that for a polynomial \( a_n x^n + a_{n-1}x^{n-1} + ... + a_0 \), if \( \frac{p}{q} \) is a rational root, then \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \). For the polynomial \( 3x^3 + 8x^2 + 5x + 2 \), \( a_0 = 2 \) and \( a_n = 3 \).

Step 2: List Factors of the Constant Term and Leading Coefficient

The constant term \( a_0 = 2 \) has factors \( \pm 1, \pm 2 \) and the leading coefficient \( a_n = 3 \) has factors \( \pm 1, \pm 3 \).

Step 3: List All Possible Rational Roots

Using the factors from Step 2, the possible rational roots are \( \pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3} \). These are obtained by forming fractions \( \frac{p}{q} \) using every combination of factors of \( a_0 \) and \( a_n \).

Step 4: Test the Possible Rational Roots

To determine which of the possible roots are actual roots of the polynomial, we evaluate the polynomial \( 3x^3 + 8x^2 + 5x + 2 \) at each of these values until the polynomial equals zero. We find that \( x = -1 \) makes the polynomial zero: \[ 3(-1)^3 + 8(-1)^2 + 5(-1) + 2 = 0 \].

Step 5: Verify Using Graphing

Using a graphing calculator or software, graph the polynomial \( 3x^3 + 8x^2 + 5x + 2 \) in the given viewing rectangle \([-3,3] \) by \([-10,10] \). Observe the points where the curve crosses the x-axis. The graph shows that the polynomial crosses at \( x = -1 \), which verifies that it is the only rational root visible in the specified range.

Key Concepts

Polynomial Roots

In mathematics, polynomial roots are the values of the variable that make a polynomial equation equal to zero. If you have an equation like \( f(x) = 3x^3 + 8x^2 + 5x + 2 \), the roots are the values of \( x \) that satisfy \( f(x) = 0 \).

These roots can be real or complex numbers, and finding them is crucial because they provide key insights into the behavior of the polynomial.

• Real Roots: These are values that intersect the x-axis on a graph. In our given polynomial, \( x = -1 \) is a real root.
• Complex Roots: Usually occur in conjugate pairs and do not intersect the x-axis; instead, they appear in the coordinate system using a complex plane.

Understanding the type and number of roots a polynomial can have helps in analyzing and graphing the function.

Rational Roots

Rational roots, sometimes referred to as rational zeros, are specifically the roots of a polynomial that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \). These roots are easier to manage and solve compared to their irrational or complex counterparts.

To find possible rational roots, we use the Rational Zeros Theorem. For a polynomial \( a_n x^n + a_{n-1}x^{n-1} + ... + a_0 \), the theorem states that any rational solution \( \frac{p}{q} \), where:

• \( p \) is a factor of the constant term \( a_0 \)
• \( q \) is a factor of the leading coefficient \( a_n \)

By forming fractions using these factors, we list all possible rational roots to check which satisfy the equation. In the exercise, after testing the possible rational roots, \( x = -1 \) was confirmed as a valid root by plugging it back into the polynomial.

Graphing Polynomial Functions

Graphing polynomial functions is an effective way to visualize the behavior of the polynomial, including identifying the roots. With the polynomial \( 3x^3 + 8x^2 + 5x + 2 \), graphing helps in confirming which of the calculated rational roots are actual roots by observing where the graph crosses the x-axis.

To start graphing:

• Identify the intercepts: Calculate where the graph meets the axes.
• Note the degree: The highest power of \( x \), which in this case is 3, suggests the shape and number of directional changes.
• Evaluate end behavior: Determines graph's direction as \( x \to \infty \) or \( -\infty \).

Using graphing software or a graphing calculator within a defined window, viewing range can highlight exact points of intersection. For our given polynomial, when viewed within the rectangle \([-3,3] \) by \([-10,10] \), we see that the only x-axis crossing, confirming the root, is at \( x = -1 \). This graphical approach helps verify results from algebraic solutions.