Problem 82 - 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. $$P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24$$

Step-by-Step Solution

Answer

Rational zeros: \(1, 2, 3\). Irrational zeros: \( \frac{1 \pm \sqrt{17}}{2} \).

Step 1: List possible rational zeros

Using the Rational Zeros Theorem, consider the factors of the constant term (24) and the leading coefficient (1). Possible rational zeros are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \).

Step 2: Test possible rational zeros using synthetic division

Test each candidate zero from Step 1 using synthetic division or direct substitution into \( P(x) \) to see which make the polynomial equal to zero. Start with \( x = 1 \).

Step 3: Confirm a rational zero

Substituting \( x = 1 \) into \( P(x) = 1 - 7 + 9 + 23 - 50 + 24 = 0 \), confirm that \( x = 1 \) is a zero. Use synthetic division on \( P(x) \) to divide by \( x-1 \), resulting in \( x^4 - 6x^3 + 3x^2 + 26x - 24 \).

Step 4: Repeat for other potential zeros

Continue synthetic division with remaining zeros and polynomial divisions. Upon use of \( x=2 \), \( P(2) = 0 \), resulting in \( x^3 - 4x^2 - 6x + 12 \).

Step 5: Simplify and find remaining rational zeros

Simplify \( x^3 - 4x^2 - 6x + 12 \) further with \( x = 3 \), another zero. Division gives \( x^2 - x - 4 \).

Step 6: Find irrational zeros via quadratic formula

For \( x^2 - x - 4 \), use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a=1, b=-1, c=-4 \). Solve to obtain \( x = \frac{1 \pm \sqrt{17}}{2} \), which are irrational zeros.

Key Concepts

Polynomial Division

Polynomial division is a process similar to long division in arithmetic, except it is applied to polynomials. This technique helps in breaking down a complex polynomial into smaller, more manageable parts, called factors. You typically use polynomial division when you suspect that a polynomial has a certain factor, or when you know one of its roots (zeros). Here’s how it works:

• Write the dividend (the polynomial you are dividing) and the divisor (the polynomial you are dividing by).
• Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this first term of the quotient, and subtract this product from the dividend.
• Repeat this process with the resulting polynomial until the degree of the remainder is less than the degree of the divisor.

Polynomial division helps simplify polynomials and can be instrumental in finding polynomial roots. It's a vital skill when dealing with higher-degree polynomials, such as in the context of finding zeros using methods like the Rational Zeros Theorem.

Synthetic Division

Synthetic Division is a shorthand method of polynomial division, particularly useful when the divisor is a linear polynomial of the form \(x - c\). This technique is faster and requires less writing than long division, making it a staple in algebra for evaluating polynomials at given points.Here's a quick guide to using synthetic division:

• Write down the coefficients of the polynomial you want to divide.
• Use the root \(c\) from the divisor \(x - c\) as a reference point.
• Bring down the leading coefficient as it is. Multiply this coefficient by \(c\), place it beneath the next coefficient, and add.
• Repeat the multiply and add steps using the result of each addition as the new number to compute with \(c\).
• The final row of numbers gives the coefficients of the quotient, with any final number representing the remainder.

This method is not only quicker for checking rational zeros but is also crucial for dividing polynomials once a zero is confirmed, helping in simplifying the polynomial further and finding additional roots.

Quadratic Formula

The quadratic formula is a universal method for finding the roots (or zeros) of a quadratic polynomial, which is any polynomial of the form \(ax^2 + bx + c = 0\). If a polynomial is reduced to a quadratic form (degree 2), this method guarantees that you can find its zeros, whether they are rational, irrational, or complex.The formula is expressed as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• \(b^2 - 4ac\) is known as the discriminant. It determines the nature of the roots.
• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root (a double root).
• A negative discriminant indicates the roots are complex (they include imaginary numbers).

In the context of the original polynomial problem, once the polynomial is reduced to \(x^2 - x - 4\), the quadratic formula is used to find the irrational zeros \(x = \frac{1 \pm \sqrt{17}}{2}\). This showcases its power when dealing with polynomials that are not easily factored by other means.

Descartes' Rule of Signs

Descartes' Rule of Signs is a useful theorem that provides insight into the number of positive and negative real roots of a polynomial. It doesn't pinpoint the exact roots but offers a clever way to estimate their count.Here's how it works:

• Count the number of sign changes in the coefficients of the polynomial when arranged in standard form (highest to lowest degree).
• The number of positive real roots is equal to the number of sign changes, or less than it by an even number.
• To find the possible number of negative real roots, replace \(x\) by \(-x\) in the polynomial and count the sign changes again.

For example, applying Descartes' Rule of Signs to our original polynomial \(P(x) = x^5 - 7x^4 + 9x^3 + 23x^2 - 50x + 24\), you can find out how many positive and negative solutions to expect, which is valuable information guiding further testing and calculation steps like synthetic division.