Problem 84 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6 $$

Step-by-Step Solution

Answer

Rational zeros: 1, \(\frac{1}{2}\), 3. Irrational zeros: \(\frac{-11 \pm \sqrt{137}}{8}\).

Step 1: Calculate Possible Rational Zeros

According to the Rational Zeros Theorem, any rational zero, \( \frac{p}{q} \), of the polynomial \( P(x) = 8x^5 - 14x^4 - 22x^3 + 57x^2 - 35x + 6 \) is such that \( p \) divides the constant term (6), and \( q \) divides the leading coefficient (8). The divisors of 6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \), and the divisors of 8 are \( \pm 1, \pm 2, \pm 4, \pm 8 \). This results in possible rational zeros: \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 2, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8}, \pm 6, \pm \frac{6}{2}, \pm \frac{6}{4}, \pm \frac{6}{8} \). Simplifying gives possible rational zeros of \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8} \).

Step 2: Test Possible Rational Zeros

Use synthetic division or direct substitution to test the possible rational zeros. Start with \( x = 1 \) and proceed through the list. After testing several, you'll find that \( x = 1 \) is a root of \( P(x) \). Thus, \( (x-1) \) is a factor of the polynomial. Use synthetic division to divide \( P(x) \) by \( (x - 1) \), and the quotient is \( Q(x) = 8x^4 - 6x^3 - 28x^2 + 29x - 6 \).

Step 3: Repeat Testing for Remaining Polynomial

With \( Q(x) \), a quartic polynomial, test the remaining possible rational zeros. After testing a few, \( x = \frac{1}{2} \) is found to be a zero of \( Q(x) \). Hence, \( (2x-1) \) is a factor. Use synthetic division again to divide \( Q(x) \) by \( (2x - 1) \), yielding a quotient \( R(x) = 8x^3 - 2x^2 - 30x + 6 \).

Step 4: Factor via the Quadratic Formula

Continue factoring \( R(x) \). Use synthetic division to check the rational zeros of \( R(x) \) using the previously listed candidates. After trials, \( x = 3 \) is found to be a zero. Divide by \( (x - 3) \), leading to a quadratic \( S(x) = 8x^2 + 22x - 2 \). With this quadratic, use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the zeros. Here, \( a = 8 \), \( b = 22 \), \( c = -2 \). Substitute these values into the quadratic formula.

Step 5: Find Final Zeros of the Quadratic

Calculate \( b^2 - 4ac = 22^2 - 4 \times 8 \times (-2) = 484 + 64 = 548 \). Thus, the zeros of \( S(x) \) are \( x = \frac{-22 \pm \sqrt{548}}{16} \). Simplify further: \( x = \frac{-22 \pm \sqrt{4 \times 137}}{16} = \frac{-22 \pm 2\sqrt{137}}{16} = \frac{-11 \pm \sqrt{137}}{8} \). These are the irrational zeros of the original polynomial \( P(x) \).

Key Concepts

Polynomial Roots

Understanding polynomial roots is essential for solving polynomial equations. A root, or zero, of a polynomial, is a value for which the polynomial equals zero. In simpler terms, if we plug in a root into the polynomial equation, the result should be zero.
Every polynomial of degree n can have up to n roots. These roots can be real or complex and multiple roots are possible. Finding these roots is crucial for graphing the polynomial function and understanding its behavior.
For polynomials with rational coefficients, the Rational Zeros Theorem helps in predicting possible rational roots, while other methods like the Quadratic Formula assist in finding both real and complex roots of lower degree polynomials.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form \((x - c)\), where \(c\) is a constant. It is much easier than the traditional long division method, especially when dealing with polynomials.
To perform synthetic division, place the constant \(c\) on the left and list the coefficients of the polynomial in descending order of their powers of \(x\). Then, carry out the operations: bring down the first coefficient, multiply it by \(c\), and add it to the next coefficient. Repeat this process through the list. The last number you get is the remainder, and the other numbers are the coefficients of the quotient polynomial.
Synthetic division is particularly useful when the goal is to determine whether a given value is a root of the polynomial and simplifies the process of checking multiple potential roots.

Quadratic Formula

The quadratic formula is a straightforward way to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) allows us to find the solutions directly.
Key to using the quadratic formula is the discriminant, \(b^2 - 4ac\). This value tells us whether the solutions are real and distinct, real and repeated, or complex:

• A positive discriminant gives two distinct real solutions.
• A zero discriminant results in one real repeated solution.
• A negative discriminant leads to two complex solutions.

The quadratic formula is indispensable for solving quadratic equations particularly when factoring is not straightforward.

Upper and Lower Bounds Theorem

The Upper and Lower Bounds Theorem is a useful tool for determining the range in which all real roots of a polynomial lie. It provides limits within which you can expect to find the roots, thus narrowing down the potential candidates.
The method involves testing possible roots using synthetic division. For the upper bound, all the results in the synthetic division process must be non-negative, while for the lower bound, they alternate in sign.
By setting these bounds, mathematicians can more efficiently locate real roots and focus their search, significantly reducing the number of values tested. This theorem is especially helpful when dealing with polynomials of higher degrees where potential rational roots are numerous.