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Chapter 3: Problem 34

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{3}-3 x^{2}-2 x+3$$

Short Answer

Expert verified
The rational zeros are 1, \( \frac{3}{2} \), and -1, and the factored form is \( 2(x-1)(2x-3)(x+1) \).

Step by step solution

01

Identify Possible Rational Zeros

According to the Rational Root Theorem, the possible rational zeros of a polynomial are the factors of the constant term divided by the factors of the leading coefficient. For the polynomial \( P(x) = 2x^{3} - 3x^{2} - 2x + 3 \), the constant term is 3, and the leading coefficient is 2. Thus, the possible rational zeros are \( \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2} \).
02

Evaluate Possible Zeros Using Synthetic Division

To find the actual rational zeros, we use synthetic division to test the possible zeros. We start with \( x = 1 \). Synthetic division of \( P(x) \) by \( x - 1 \) yields a remainder of 0, indicating that \( x = 1 \) is a zero. Repeat for \( x = -1 \), \( x = 3 \), etc. and find that \( x = \frac{3}{2} \) is another zero.
03

Factorize the Polynomial

With known zeros \( x = 1 \) and \( x = \frac{3}{2} \), we form the factors \( (x - 1) \) and \( (2x - 3) \). Given \( P(x) \) is a cubic polynomial, it will have three roots, which means it can be expressed as \( P(x) = 2(x-1)(2x-3)Q(x) \). Calculate \( Q(x) \) to find the remaining factorisation. By division, find \( Q(x) = (x + 1) \).
04

Write the Polynomial in Factored Form

Combining the factors found, the polynomial can be completely factored as \( P(x) = 2(x - 1)(2x - 3)(x + 1) \). This shows all rational zeros \( x = 1, \ x = \frac{3}{2}, \ -1 \).

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Key Concepts

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

Polynomial Factorization
Polynomial factorization involves breaking down a complex polynomial into a product of simpler polynomials. This helps in understanding the polynomial's structure and finding its roots. When you perform polynomial factorization, you are expressing the original polynomial as a multiplication of its factors.
  • Start by identifying the polynomial equation. In this case: \( P(x) = 2x^3 - 3x^2 - 2x + 3 \).
  • Determine the polynomial's roots, as these will help you form the factors. The roots here are \( x=1 \), \( x=\frac{3}{2} \), and \( x=-1 \).
  • Write these roots as factors: \( x-1 \), \( 2x-3 \), and \( x+1 \).
  • Finally, multiply these factors to check that they expand back to the original polynomial.
The factorization simplifies to \( 2(x - 1)(2x - 3)(x + 1) \). This process is crucial in solving polynomial equations and interpreting their graphical representations.
Synthetic Division
Synthetic division is a simplified form of polynomial division, used to find roots or zeroes of polynomials quickly. It uses less complicated arithmetic and focuses on coefficients rather than variables. To use synthetic division:
  • Take one possible zero, say \( x = 1 \), and use it in a synthetic division setup with your polynomial's coefficients: \( 2, -3, -2, 3 \).
  • Place the possible zero on the left and set up the row of coefficients on the right.
  • Bring down the first coefficient. Multiply it by the zero and add to the next coefficient. This is repeated across the row.
  • If the result at the last coefficient position is zero, the tested value is indeed a root.
In our case, synthetic division verified that \( x=1 \) and \( x=\frac{3}{2} \) are roots. This method is preferable for its speed and efficiency, especially with higher-degree polynomials.
Rational Zeros
Rational zeros are the solutions to the polynomial equation that can be expressed as fractions. According to the Rational Root Theorem, if there's a rational zero \( \frac{p}{q} \), then \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
  • Identify potential rational zeros by calculating these factors. For \( P(x) = 2x^3 - 3x^2 - 2x + 3 \), the constant term is 3, and the leading coefficient is 2.
  • Possible zeros are \( \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2} \).
  • Test these using synthetic division to see if they actually satisfy the polynomial equation (resulting in a remainder of zero).
  • This can greatly simplify the process of finding all zeros of a polynomial without extensive guesswork.
In this exercise, applying this theorem helped identify \( x = 1 \), \( x = \frac{3}{2} \), and \( x = -1 \) as rational zeros, which were then used to factor the polynomial.

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Most popular questions from this chapter

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