Problem 31 - College Algebra

Question

For the following exercises, use the Rational Zero Theorem to find all real zeros. \(2 x^{3}-5 x^{2}+9 x-9=0\)

Step-by-Step Solution

Answer

The real zero of the polynomial is \(x = 1\).

Step 1: Introduction to the Rational Zero Theorem

The Rational Zero Theorem helps us find possible rational roots of a polynomial equation of the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\). According to the theorem, any rational solution \(\frac{p}{q}\) is such that \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\).

Step 2: Identify Coefficients and Apply Theorem

Identify the coefficients in the polynomial: constantly term \(a_0 = -9\) and leading coefficient \(a_n = 2\). The factors of \(-9\) are \(\pm 1, \pm 3, \pm 9\) and the factors of \(2\) are \(\pm 1, \pm 2\). Possible rational zeros are fractions formed with these factors: \(\pm 1, \pm \frac{1}{2}, \pm 3, \pm \frac{3}{2}, \pm 9, \pm \frac{9}{2}\).

Step 3: Test Possible Rational Zeros

Test each possible rational zero by substituting \(x\) in polynomial \(2x^3 - 5x^2 + 9x - 9\) and checking which value gives zero. Upon testing, we find \(x = 1\) as a root because substituting yields zero: \(2(1)^3 - 5(1)^2 + 9(1) - 9 = 0\).

Step 4: Polynomial Division

With \(x=1\) confirmed as a root, perform synthetic division to divide the polynomial by \(x-1\). The result is a quadratic equation \(2x^2 - 3x + 9\).

Step 5: Solve the Quadratic Equation

Solve the quadratic equation \(2x^2 - 3x + 9 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = -3\), \(c = 9\). This yields no real solution since the discriminant \(b^2 - 4ac = (-3)^2 - 4(2)(9) = 9 - 72 = -63\) is negative, indicating complex roots.

Step 6: Conclusion

The only real zero of the polynomial \(2x^3 - 5x^2 + 9x - 9\) is \(x = 1\). The quadratic factor \(2x^2 - 3x + 9\) has no real roots.

Key Concepts

Polynomial Equation

A polynomial equation is an expression that can be expressed in the form of a sum of terms consisting of a variable raised to a power and multiplied by coefficients. The general form for a polynomial equation of degree n is:

• \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0\)

Each term consists of a coefficient (\(a_n\)) and the variable (\(x\)) raised to a non-negative integer power.
In our exercise, the polynomial equation is \(2x^3 - 5x^2 + 9x - 9 = 0\).
To find the roots of a polynomial, several methods can be used, one of which is applying the Rational Zero Theorem. This helps in identifying possible rational roots by considering the factors of the constant and leading coefficient. Techniques such as synthetic division and the quadratic formula can further aid in refining these possible solutions.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear polynomial of the form \(x-c\). It is particularly useful in determining whether a potential root is an actual root of the polynomial. This method involves writing only the coefficients of the polynomial and performing specific operations to simplify the division process.
In our example, after identifying \(x=1\) as a rational root using the Rational Zero Theorem, synthetic division is used to divide the polynomial \(2x^3 - 5x^2 + 9x - 9\) by \(x-1\).

• Arrange the coefficients: \([2, -5, 9, -9]\).
• Use \(c = 1\) in synthetic division.
• Perform synthetic division by listing the operation results, ultimately arriving at \(2x^2 - 3x + 9\).

The result of the synthetic division is a quadratic polynomial, which can be further analyzed for additional roots. This step is key because it reduces the degree of the polynomial, making it easier to solve.

Quadratic Formula

The quadratic formula is a tool used to find the roots of a quadratic equation, which is any equation of the form \(ax^2 + bx + c = 0\). It provides a complete solution to any quadratic equation, even if the roots are complex. The formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. The term under the square root, \(b^2 - 4ac\), is called the discriminant. It tells us about the nature of the roots.

• If the discriminant is positive, there are two real roots.
• If it is zero, there is one real root.
• If negative, the roots are complex (non-real).

In our solution, we apply the quadratic formula to \(2x^2 - 3x + 9 = 0\) resulting from synthetic division. The discriminant here is \(-63\), which is negative, indicating that the quadratic has no real roots, only complex ones.