Problem 81 - 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. $$ P(x)=4 x^{4}-21 x^{2}+5 $$

Step-by-Step Solution

Answer

The rational zeros are \(x = 1, -1, \frac{1}{2}, -\frac{1}{2}\). No irrational zeros exist.

Step 1: Identify Possible Rational Zeros

According to the Rational Zeros Theorem, any rational zero of the polynomial \(P(x) = 4x^4 - 21x^2 + 5\) is a factor of the constant term (5) divided by a factor of the leading coefficient (4). Thus, the possible rational zeros are \(\pm 1, \pm 5, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{4}, \pm \frac{5}{4}\).

Step 2: Test Possible Zeros using Synthetic Division

Test each possible rational zero by substituting it into the polynomial या by using synthetic division to see if the remainder is zero. After testing, we find that \(x = 1\) and \(x = -1\) are zeros of the polynomial.

Step 3: Factor the Polynomial using Found Zeros

Since \(x = 1\) and \(x = -1\) are zeros, we can express \(P(x)\) as \((x - 1)(x + 1)(Ax^2 + Bx + C)\). Compute and simplify to find \(Ax^2 + Bx + C\). We get \(P(x) = (x^2 - 1)(4x^2 - 1)\).

Step 4: Factor the Remaining Quadratic Expressions

Recognize \(x^2 - 1 = (x - 1)(x + 1)\) and \(4x^2 - 1 = (2x - 1)(2x + 1)\). Thus, the polynomial \(P(x)\) is factored as \((x - 1)^2(x + 1)^2(2x - 1)(2x + 1)\).

Step 5: Determine All Zeros

The zeros are derived from the factors as \(x = 1, -1, \frac{1}{2}, -\frac{1}{2}\), and all are rational.

Key Concepts

Upper and Lower Bounds Theorem

The Upper and Lower Bounds Theorem helps us in identifying the range within which the real zeros of the polynomial lie. By using synthetic division, we can identify numbers that serve as upper or lower bounds. Here’s how it works:

• An upper bound is a number where all resulting numbers in the synthetic division row are non-negative. This tells us that there are no zeros larger than this number.
• A lower bound is a number where the resulting row after synthetic division alternates between non-positive and non-negative numbers, indicating no zeros smaller than this number.

By leveraging this theorem, we can narrow down the possible values for zeros, streamlining our search and verifying the zeros found through testing.

Descartes' Rule of Signs

Descartes' Rule of Signs is a handy technique to determine the number of positive and negative real zeros. Here’s how it breaks down:

• To find the number of possible positive real zeros, count the number of sign changes in the polynomial's terms.
• For negative real zeros, substitute \(x\ ightarrow -x\) and then count the sign changes.

The counts give the possible number of positive and negative real roots. However, these counts aren’t definite, as they could be fewer by a factor of 2 (e.g., if you count 3 sign changes, you may have 1 or 3 positive roots). Using this rule, we identify how many zeros to explore during testing with synthetic division.

Synthetic Division

Synthetic division is a streamlined method of polynomial division, useful especially for finding zeros and testing possible rational zeros from the Rational Zeros Theorem. Here’s the process:

• Choose a potential zero and write it to the left of a vertical bar.
• List the coefficients of the polynomial to the right.
• Bring down the first coefficient to the row below.
• Multiply it by the potential zero, writing the result under the next coefficient, then add downwards.

Repeat this multiply-add process across the row. If you end up with a remainder of zero, it confirms the potential zero is an actual root. This method is faster than long division and highly effective in narrowing down actual zeros.

Quadratic Formula

The Quadratic Formula is a crucial tool for finding the zeros of quadratic polynomials where factoring seems complex or unclear. The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula can find both real and complex zeros. It’s derived from the standard form of a quadratic equation, \(ax^2 + bx + c = 0\). Here’s when you use it:

• When you have reduced a polynomial to a quadratic form, and simple factorization is not possible.
• If the discriminant, \(b^2 - 4ac\), is positive, you get two real solutions.
• If zero, you find one real solution (a repeated zero).
• If negative, the zeros are complex.

In our exercise, after identifying all factors, the polynomial factored cleanly, so using the Quadratic Formula was not necessary. However, it is an indispensable tool for more complicated expressions.