Problem 73 - 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. $$ P(x)=2 x^{4}+3 x^{3}-4 x^{2}-3 x+2 $$

Step-by-Step Solution

Answer

Rational zeros: 1, -1. Complex zeros: \( i\sqrt{\frac{3}{2}}, -i\sqrt{\frac{3}{2}} \).

Step 1: Identify Potential Rational Zeros

According to the Rational Zeros Theorem, any possible rational zero of the polynomial \( P(x) = 2x^4 + 3x^3 - 4x^2 - 3x + 2 \) is a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term 2 and \( q \) is a factor of the leading coefficient 2. Therefore, the possible rational zeros are the factors of 2: \( \pm 1, \pm 2 \). Divide each by the factors of 2: \( \pm 1, \pm \frac{1}{2}, \pm 2, \pm \frac{2}{2} \). After simplification, the potential rational zeros are: \( \pm 1, \pm \frac{1}{2}, \pm 2 \).

Step 2: Synthetic Division to Test Zeros

To determine which, if any, of the potential rational zeros are actual zeros of the polynomial, apply synthetic division. Start by testing \( x = 1 \): - Coefficients: \( 2, 3, -4, -3, 2 \).- Perform synthetic division. - The remainder is 0, indicating \( x = 1 \) is a zero.Repeat for other potential candidates if needed, until all zeros are identified. After testing, \( x = -1 \) is also found to be a zero. With these, factor the polynomial as: \( (x-1)(x+1)Q(x) \).

Step 3: Factor the Polynomial

Given that \( x = 1 \) and \( x = -1 \) are zeros, the polynomial can be divided: \[ P(x) = (x-1)(x+1)Q(x) \] Use polynomial division to find \( Q(x) \) dividing by \((x^2-1)\). The quotient \( Q(x) = 2x^2 + 3 \).

Step 4: Solve Remaining Quadratic Equation

The reduced polynomial is \( 2x^2 + 3 \). Solve this quadratic for zeros:\[ 2x^2 + 3 = 0 \]Subtract 3 from both sides: \( 2x^2 = -3 \).Divide by 2: \( x^2 = -\frac{3}{2} \).Take square roots: \( x = \pm i\sqrt{\frac{3}{2}} \).

Step 5: Determine all Zeros

Combine all zeros from the analysis:- Rational zeros from steps: \( x = 1, -1 \).- Irrational zeros derived are actually complex: \( x = i\sqrt{\frac{3}{2}}, -i\sqrt{\frac{3}{2}} \). The polynomial has no other real roots or irrational roots in the sense of being rationally irrational (i.e., non-complex).

Key Concepts

Rational Zeros Theorem

The Rational Zeros Theorem is a powerful tool that helps us identify possible rational zeros of a polynomial. It states that any potential rational zero of a polynomial expressed as \( P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \) will be of the form \( \frac{p}{q} \). Here, \( p \) is a factor of the constant term \( a_0 \) and \( q \) is a factor of the leading coefficient \( a_n \).

To apply this theorem to the polynomial \( P(x) = 2x^4 + 3x^3 - 4x^2 - 3x + 2 \), we begin by identifying the factors of the constant term and the leading coefficient. The constant term is 2, with factors \( \pm 1, \pm 2 \). The leading coefficient is also 2, with the same factors.

This results in potential rational zeros of \( \pm 1, \pm \frac{1}{2}, \pm 2 \). By identifying these values, we have a good starting point to test and find actual zeros using synthetic division or other methods.

Descartes’ Rule of Signs

Descartes’ Rule of Signs provides a quick way to estimate the number of positive and negative real zeros in a polynomial. It involves counting the number of sign changes in a polynomial’s coefficients. For example, for \( P(x) = 2x^4 + 3x^3 - 4x^2 - 3x + 2 \), we analyze the sequence of coefficients: \( 2, 3, -4, -3, 2 \).

Examining the changes:

• 2 to 3: no sign change
• 3 to -4: sign change
• -4 to -3: no sign change
• -3 to 2: sign change

Thus, there are two sign changes, implying up to 2 positive real zeros, either 2 or 0, depending on the behavior inside the intervals.

To find negative roots, substitute \( -x \) into the polynomial, resulting in \( 2(-x)^4 + 3(-x)^3 - 4(-x)^2 - 3(-x) + 2 \) and re-evaluate the signs. Using Descartes’ Rule of Signs is useful for narrowing down where the zeros might occur, which can guide further zero-finding techniques like synthetic division or factoring.

Synthetic Division

Synthetic division is an efficient method for dividing polynomials, especially when determining if a certain value is a zero of the polynomial. Rather than performing full polynomial long division, synthetic division simplifies the process using only the coefficients.

To use synthetic division on \( P(x) = 2x^4 + 3x^3 - 4x^2 - 3x + 2 \), choose a potential zero, such as \( x = 1 \), and list the coefficients: 2, 3, -4, -3, 2.

Perform the operations:

• Write down the leading coefficient (2) in the bottom row.
• Multiply it by the potential zero (1) and add to the next coefficient (3).
• Repeat this process for each coefficient.

Upon reaching a remainder of zero indicates that 1 is indeed a zero of the polynomial. If the remainder is non-zero, then \( x = 1 \) is not a zero. Repeat the test with other potential rational zeros identified earlier until all roots are found.

Quadratic Formula

When the polynomial has been reduced to a simpler quadratic form, such as in this exercise where division has led to the expression \( Q(x) = 2x^2 + 3 \), the quadratic formula is quite handy. This formula lets us solve quadratics of the form \( ax^2 + bx + c = 0 \).

The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In our example, for the quadratic \( 2x^2 + 3 = 0 \), we assign \( a = 2 \), \( b = 0 \), and \( c = 3 \).

Plug these into the formula:

• The discriminant \( b^2 - 4ac = 0 - 4(2)(3) = -24 \), which is negative.
• Thus, the polynomial has no real zeros, but rather complex ones.
• Solving gives \( x = \pm i\sqrt{\frac{3}{2}} \), indicating complex irrational roots.

The quadratic formula is incredibly useful when faced with irreducible quadratics, particularly when identifying complex roots.