Problem 80 - Precalculus Mathematics for Calculus

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' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=2 x^{4}+15 x^{3}+31 x^{2}+20 x+4$$

Step-by-Step Solution

Answer

Rational zero: x = -1 (with multiplicity 2); Irrational zeros: \(x = \frac{-11 + \sqrt{89}}{4}\) and \(x = \frac{-11 - \sqrt{89}}{4}\).

Step 1: Apply Rational Zeros Theorem

The Rational Zeros Theorem suggests that possible rational zeros of a polynomial are the ratios of the factors of the constant term (4) to the factors of the leading coefficient (2). The factors of 4 are ±1, ±2, ±4 and the factors of 2 are ±1, ±2. Thus, possible rational zeros are ±1, ±2, ±4, ±1/2, ±2.

Step 2: Testing possible rational zeros using synthetic division

Test each of the possible rational zeros using synthetic division. If the remainder is zero, then it is a rational zero: - Test x = -1: The remainder is 0. Hence, x = -1 is a rational zero. - Test other values: None yield zero remainder.

Step 3: Perform Polynomial Division

Since x = -1 is a root, divide \(P(x)\) by \(x + 1\) using synthetic or long division:Dividing by \(x + 1\) gives us another polynomial: \(2x^3 + 13x^2 + 18x + 4\).

Step 4: Repeat Rational Zeros Theorem for Remaining Polynomial

Now, apply the Rational Zeros Theorem to the cubic polynomial \(2x^3 + 13x^2 + 18x + 4\): Possible rational zeros are ±1, ±2, ±4, ±1/2, ±2 (as before).

Step 5: Test Remaining Possible Rational Zeros

Test these zeros using synthetic division: - Test x = -1: The remainder is again 0. Hence, x = -1 is a rational zero again. - Test other values: None yield zero remainder again.

Step 6: Further Factor the Polynomial

Divide \(2x^3 + 13x^2 + 18x + 4\) by \(x + 1\) again to obtain \(2x^2 + 11x + 4\).

Step 7: Solve Quadratic Equation for Remaining Polynomial

Solve \(2x^2 + 11x + 4 = 0\) using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substituting a = 2, b = 11, c = 4:\[x = \frac{-11 \pm \sqrt{121 - 32}}{4}\]\[x = \frac{-11 \pm \sqrt{89}}{4}\]These are irrational zeros: \(x = \frac{-11 + \sqrt{89}}{4}\) and \(x = \frac{-11 - \sqrt{89}}{4}\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials similar to how you divide numbers. When you divide a polynomial by another, you are seeking to rewrite it as a product of a divisor multiplied by a quotient, plus a remainder. This technique is often used when you know one of the polynomial's roots, allowing you to factor the polynomial more easily.

You might either use long division or synthetic division to perform polynomial division. Long division is similar to the process you use with numbers, involving subtraction of multiples of the divisor and bringing down the next term in each step. It can be quite detailed.

• Steps: Align the polynomials in decreasing degree order.
• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by the result, and subtract it from the dividend.
• Repeat the process with the new polynomial that results from the subtraction.

This remains a useful technique especially if it reveals further opportunities to factor the polynomial or identify additional zeros.

Quadratic Formula

The quadratic formula is a reliable way to find the zeros of any quadratic equation, which is any polynomial in the form ax^2 + bx + c = 0. It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To use this formula:

• Identify the coefficients (a, b, and c) from your quadratic equation.
• Calculate the discriminant, \(b^2 - 4ac\), to determine the nature of the roots.
• Substitute these values into the quadratic formula to find the zero(s).
• The "plus-minus" sign (±) indicates that there are typically two solutions.

For the polynomial from the original exercise, the quadratic formula helped find the irrational zeros. When the discriminant, which dictates the nature of the roots, is not a perfect square, the results are irrational numbers, as demonstrated in the solution above.

Descartes' Rule of Signs

Descartes' Rule of Signs is a method used to estimate the number of positive and negative real roots of a polynomial. It helps predict the behavior of a polynomial's roots by examining the signs of its coefficients.

When applied:

• Count how many times the sign changes between consecutive non-zero coefficients in the polynomial expression. This count will give you the maximum number of positive real zeros.
• To find potential negative real roots, replace every instance of x with -x and recount the sign changes.
• It provides the maximum possible number of roots which is useful for guessing and checking potential zeros.

Remember that the actual number of positive or negative zeros may be fewer than predicted, typically reduced by an even number. In this exercise, Descartes' Rule can assist by narrowing down or confirming the potential number of zero sign changes that need exploring with the Rational Zeros Theorem.

Synthetic Division

Synthetic division is a quick shorthand method to divide polynomials, especially useful when you need to test possible roots derived from the Rational Zeros Theorem. This method simplifies Polynomial Division, often requiring fewer calculations and less space compared to long division.

Here's how synthetic division works:

• Write down the coefficients of the polynomial.
• Use the potential zero (from the Rational Zeros Theorem) to start the synthetic division process.
• Bring down the first coefficient as your initial lead.
• Multiply this lead with the potential zero and add it to the next coefficient, continuing the same pattern until complete.
• If the result, or remainder, is zero, the potential zero is indeed an actual zero of the polynomial.

In the given exercise, testing potential zeros with synthetic division revealed actual zeros, leading to further analysis and division of the polynomial.