Problem 77 - 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’ 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, 4. Irrational zeros: None.

Step 1: Identify Possible Rational Zeros

According to the Rational Zeros Theorem, any rational zero of the polynomial \( P(x) = x^5 - 7x^4 + 9x^3 + 23x^2 - 50x + 24 \) can be written as \( \frac{p}{q} \), where \( p \) is a factor of the constant term (24) and \( q \) is a factor of the leading coefficient (1). The factors of 24 are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \), so the possible rational zeros could be any of these.

Step 2: Test Possible Rational Zeros

Use synthetic division or direct substitution to test each possible rational zero from Step 1. Start with \( x = 1 \): If we use synthetic division or direct substitution: \[1^5 - 7(1)^4 + 9(1)^3 + 23(1)^2 - 50(1) + 24 = 0\]Check each potential zero similarly to find which ones are actually zeros of the polynomial.

Step 3: Find Factors of the Polynomial

After finding a real root, break the polynomial into a linear factor and a quartic polynomial. For example, if \( x = 1 \) is a zero, one factor is \( x - 1 \), and the quotient is \( x^4 - 6x^3 + 3x^2 + 26x - 24 \). Factor this further as needed or continue finding other rational roots.

Step 4: Use the Quadratic Formula for Remaining Irrational Zeros

If the lower degree polynomial can further be reduced to a quadratic form that doesn't factor neatly, apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the remaining zeros. For example, solve \( x^2 - 7x + 12 = 0 \) to find irrational roots.

Key Concepts

Polynomial Factoring

Factoring a polynomial means expressing it as a product of simpler polynomials. It is like breaking down a large number into its prime factors. For the given polynomial \( P(x) = x^{5} - 7x^{4} + 9x^{3} + 23x^{2} - 50x + 24 \), the goal is to express it as a product of lower-degree polynomials. Knowing the rational zeros helps in pinpointing these factors.
One common method is to test for zeros using the Rational Zeros Theorem, and upon finding a zero, say \( x = a \), we can say \((x - a)\) is a factor of the polynomial. This process involves creating a smaller polynomial, or quotient, which can be further factored until it's only irreducible factors remain. Each successful factorization simplifies the polynomial and brings us closer to its complete decomposition.

Descartes’ Rule of Signs

Descartes' Rule of Signs offers insight into the number of positive and negative real zeros a polynomial might have based on the number of sign changes in its terms. In this rule, be sure to look at the coefficients:

• For positive real zeros, count the sign changes in the polynomial \( P(x) \).
• For negative real zeros, count the sign changes in \( P(-x) \).

In our polynomial example, \( x^{5} - 7x^{4} + 9x^{3} + 23x^{2} - 50x + 24 \), the sequence of signs goes: +, -, +, +, -, +, with three sign changes, suggesting there are 3 or 1 positive real roots. Checking \( P(-x) \) could identify potential negative roots. It won't give exact roots but helps narrow the possibilities.

Synthetic Division

Synthetic division is a simplified form of polynomial division, similar to long division but easier for polynomials. You use it to test possible zeros by seeing if there is no remainder when dividing by \((x - c)\).
This method helps identify both zeros and factors efficiently. For each potential zero, like the \( x = 1 \) tested in our example, we utilize synthetic division to check divisibility. The final value helps determine whether \((x - 1)\) is indeed a factor of \( P(x) \). This is crucial in breaking the polynomial into smaller pieces, which can be subsequently factored or solved using other methods.

Quadratic Formula

The quadratic formula is a method used to find the roots of a quadratic equation, which takes the form \( ax^2 + bx + c = 0 \). It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]When working with polynomials, you may reduce a high-degree polynomial to a quadratic form after several factorization steps, as demonstrated in our example.
It's particularly helpful when the quadratic doesn't factor easily. You simply plug in the coefficients into the formula, leading to potential real or complex solutions. Thus, the quadratic formula acts as a fail-safe for finding irrational or real roots when factoring doesn't yield an obvious result.