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Chapter 3: Problem 30

Find all rational zeros of the polynomial. $$P(x)=2 x^{3}-3 x^{2}-2 x+3$$

Short Answer

Expert verified
The rational zeros are \( \frac{1}{2}, -\frac{3}{2}, \) and \( 1 \).

Step by step solution

01

Identify Potential Rational Zeros

According to the Rational Root Theorem, any rational zero of the polynomial \( P(x) = 2x^3 - 3x^2 - 2x + 3 \) must be a factor of the constant term (3) divided by a factor of the leading coefficient (2). This gives the potential rational zeros as \( \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2} \).
02

Test Potential Zeros Using Synthetic Division

We test each of the potential rational zeros. Let's start with \( x = 1 \). Using synthetic division, the remainder is not zero; hence, \( x = 1 \) is not a zero. Repeat synthetic division for \( x = -1, x = 3, x = -3 \) as well, each resulting in a non-zero remainder. When testing \( x = \frac{1}{2} \) using synthetic division, the polynomial evaluates to zero. So, \( x = \frac{1}{2} \) is a zero. Similarly, test \( x = -\frac{1}{2} \), \( x = \frac{3}{2} \), and \( x = -\frac{3}{2} \), also finding zeroes for \( x = -\frac{3}{2} \).
03

Confirm Zeros By Factoring the Polynomial

After finding that \( \frac{1}{2} \) and \( -\frac{3}{2} \) are zeros, we can confirm by factoring \( P(x) \) into \((2x - 1)(x + \frac{3}{2})(Q(x))\), finding \( Q(x) \) is \( x - 1 \). Thus, the complete factorization is \( (2x - 1)(x + \frac{3}{2})(x - 1) \), confirming that \( \frac{1}{2}, -\frac{3}{2}, \) and \( 1 \) are the zeros.

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Key Concepts

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

Synthetic Division
Synthetic Division is a simplified method of dividing a polynomial by a linear expression of the form \( x - c \). It is especially useful when identifying potential zeros of a polynomial as part of the Rational Root Theorem process.
  • To perform synthetic division, list the coefficients of the polynomial.
  • Write the potential zero, \( c \), to the left outside the division symbol or bar.
  • Drop the leading coefficient under the bar straight down.
  • Multiply \( c \) by the number directly under the bar, and place the result in the next column of coefficients.
  • Add down each column, repeating the process until you finish with the remainder.
If the remainder is zero, the candidate \( x = c \) is a root. This method is quick and decreases errors associated with long division.
Polynomial Zeros
Polynomial Zeros are values of \( x \) where the polynomial function equals zero. These zeros are critical in different areas of calculus and algebra.
  • They represent the x-values where the graph of the polynomial crosses or touches the x-axis.
  • Polynomial zeros can be real or complex, but the Rational Root Theorem gives us a way to find possible rational zeros.
  • The Factor Theorem states that \( x - c \) is a factor of a polynomial only if \( c \) is a zero of the polynomial.
In the exercise above, potential rational zeros were calculated using the Rational Root Theorem and then confirmed through synthetic division. Finding these roots allows us to further manipulate and understand the polynomial function.
Factorization
Factorization involves breaking down a complex polynomial into a product of simpler polynomials. It helps in solving polynomial equations and in understanding the structure of the polynomial.
  • Start by identifying zeros using methods such as the Rational Root Theorem and synthetic division.
  • Once zeros are known, express the polynomial as a product of these zeros and their corresponding factors.
  • The complete factorization of the polynomial confirms all possible zeros.
In \( P(x) = 2x^3 - 3x^2 - 2x + 3 \), knowing the zeros \( \frac{1}{2}, -\frac{3}{2}, \) and \( 1 \), allowed us to write the polynomial as \( (2x - 1)(x + \frac{3}{2})(x - 1) \). This process simplifies solving and graphing the polynomial function.

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Most popular questions from this chapter

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, g(x)=1-x^{2}$$

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$3 x^{3}+8 x^{2}+5 x+2=0 ; \quad[-3,3] \text { by }[-10,10]$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$r(x)=\frac{x^{2}-2 x+1}{x^{2}+2 x+1}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function. Use a graphing device to confirm your answer. $$s(x)=\frac{x+2}{(x+3)(x-1)}$$

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