Problem 78 Find all rational zeros of the p... [FREE SOLUTION]

Chapter 3: Problem 78

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. $$P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6$$

Short Answer

Expert verified

The rational zero is $ x=1 $. Further zeros may include irrational solutions found via the quadratic formula.

Step by step solution

Apply the Rational Zeros Theorem

According to the Rational Zeros Theorem, any rational zero of the polynomial $ P(x) = 8x^5 - 14x^4 - 22x^3 + 57x^2 - 35x + 6 $ can be expressed as a fraction $ \frac{p}{q} $, where $ p $ is a factor of the constant term (6) and $ q $ is a factor of the leading coefficient (8). List the factors of 6: $ \pm 1, \pm 2, \pm 3, \pm 6 $ and factors of 8: $ \pm 1, \pm 2, \pm 4, \pm 8 $. The potential rational zeros are $ \frac{p}{q} $ using these factors, such as $ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 2, \pm 3, \pm 6, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8} $.

Test the potential rational zeros

Test each potential rational zero by substituting into $ P(x) $. We evaluate several using synthetic division: $ P(1) $ gives a remainder of 0, indicating $ x=1 $ is a zero. Also, $ P\left(\frac{1}{2}\right) $, $ P(-1) $, and $ P(3) $ are tested in a similar manner. Only $ x=1 $ results in zero remainder.

Factor the polynomial using found zero

Since $ x=1 $ is a zero, $ x-1 $ is a factor. Divide $ P(x) $ by $ x-1 $ using synthetic division or polynomial division to obtain the quotient $ 8x^4 - 6x^3 - 28x^2 + 29x - 6 $.

Repeat for quotient polynomial

Apply the Rational Zeros Theorem again or factor further. Test rational zeros for the new polynomial $ 8x^4 - 6x^3 - 28x^2 + 29x - 6 $. Repeat the testing process and possibly find additional rational zeros.

Identify irrational zeros

If the quotient polynomial becomes a quadratic or can be factored further, or if non-rational solutions are required, apply the quadratic formula on any irreducible quadratic factor resulting from division. Assume $ ax^2 + bx + c = 0 $ is such a factor, then use $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ to find irrational zeros, if any.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps us break down complex polynomials into simpler parts. For example, when we divide our original polynomial $ P(x) = 8x^5 - 14x^4 - 22x^3 + 57x^2 - 35x + 6 $ by $ x - 1 $ (because we found that $ x =1 $ is a zero), we obtain a quotient that is a polynomial of one degree less.
The process works as follows:

First, divide the leading term of the dividend by the leading term of the divisor.
Multiply the entire divisor by the result from the first step and subtract from the dividend.
Repeat with the new polynomial formed from the subtraction.

This cycle is repeated untill we obtain a remainder. A zero remainder indicates that the divisor is a factor of the original polynomial.

Synthetic Division

Synthetic division is a simplified form of polynomial division, mainly used when dividing by a linear factor of the form $ x - c $. It is quicker and easier than the traditional long division method, particularly when checking potential zeros.
To perform synthetic division for our polynomial $ P(x) $:

Write down the coefficients of the polynomial.
Use the zero of the divisor $ c $ next to the coefficients and draw a line.
Bring down the first coefficient without change.
Multiply $ c $ by this first coefficient and add it to the next coefficient.
Repeat the multiply-and-add process for all coefficients.

Finding a sum of zero means the divisor is a factor and provides a new polynomial quotient. It is a good method to verify possible rational zeros quickly.

Descartes' Rule of Signs

Descartes' Rule of Signs is a tool used for predicting how many positive and negative real roots a polynomial might have. It is based on the number of times the sign changes in the sequence of coefficients of the polynomial.
For positive roots:

Count the number of sign changes in the polynomial's coefficients.
The number of positive real roots is equal to the number of sign changes, or less than it by an even number.

For negative roots:

Substitute $ x $ with $ -x $ and count the sign changes again.

It doesn't tell you exactly how many roots there are, but it helps narrow down possibilities. Using this rule along with other methods, like synthetic division, can reveal the most likely candidates for real zeros.

Quadratic Formula

The quadratic formula is a reliable method used to find roots of quadratic equations of the form $ ax^2 + bx + c = 0 $. It's especially useful when other methods, such as factoring, synthetic division, or graphing, aren't applicable.
To apply the quadratic formula:

Plug the coefficients $ a, b, $ and $ c $ into the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Calculate the discriminant $ b^2 - 4ac $ to determine the nature of the roots.
If the discriminant is positive, there are two distinct real roots; if it's zero, exactly one real root; and if negative, the roots are complex and not real.

In the process of solving our polynomial $ P(x) $, we might reach a point where a remaining factor is a quadratic, and the quadratic formula can quickly provide the solution to any irrational roots.

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Short Answer

Step by step solution

Apply the Rational Zeros Theorem

Test the potential rational zeros

Factor the polynomial using found zero

Repeat for quotient polynomial

Identify irrational zeros

Key Concepts

Polynomial Division

Synthetic Division

Descartes' Rule of Signs

Quadratic Formula

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