Problem 34 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The rational zeros are 1, -1, and \( \frac{3}{2} \). Factorization is \( (x - 1)(x + 1)(2x - 3) \).

Step 1: Identify the Rational Root Theorem

The Rational Root Theorem tells us that any rational solution, or root, of the polynomial equation \( P(x) = 0 \) is a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term (3) and \( q \) is a factor of the leading coefficient (2). Thus, possible rational roots are \( \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2} \).

Step 2: Test Possible Rational Roots

Substitute each possible rational root from the list into the polynomial to check which coefficients make the polynomial equal zero. Start with \( x = 1 \):\[ P(1) = 2(1)^3 - 3(1)^2 - 2(1) + 3 = 0. \] The polynomial evaluates to zero, so \( x = 1 \) is a root.

Step 3: Use Synthetic Division

Use synthetic division with \( x = 1 \) to factor the polynomial. Divide \( 2x^3 - 3x^2 - 2x + 3 \) by \( x - 1 \) to find the quotient polynomial:\[\begin{array}{r|rrrr}1 & 2 & -3 & -2 & 3 \ & & 2 & -1 & -3 \ ext{---}&---&---&---&--- \ & 2 & -1 & -3 & 0 \ ext{---}\end{array}\] The quotient is \( 2x^2 - x - 3 \), thus \( P(x) = (x - 1)(2x^2 - x - 3) \).

Step 4: Factor the Quadratic

Try to factor the quadratic \( 2x^2 - x - 3 \). Look for two numbers that multiply to \( 2 \times (-3) = -6 \) and add to \( -1 \). These numbers are \( -3 \) and \( 2 \). Rewrite and factor by grouping:\[ 2x^2 - 3x + 2x - 3 = x(2x - 3) + 1(2x - 3) = (x + 1)(2x - 3). \] So, \( P(x) = (x - 1)(x + 1)(2x - 3) \).

Step 5: Verify the Factorization

Multiply the factors back to ensure they give the original polynomial:\[ (x-1)(x+1)(2x-3) = (x^2-1)(2x-3) = 2x^3 - 3x^2 - 2x + 3. \]The factorization is correct, which confirms that the rational roots are \( x = 1, -1, \frac{3}{2} \).

Key Concepts

Synthetic Division

Synthetic division is a quick method to divide a polynomial by a linear binomial of the form \( x - c \), where \( c \) is a root of the polynomial. This method is particularly useful to simplify polynomials and find other factors once a root is known.
To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial. If any terms are missing, use a zero as the coefficient.
• Bring down the leading coefficient to the bottom row.
• Multiply \( c \) by the value just written on the bottom row. Write the result in the next column on the second row.
• Add the values in the current column and write the result on the bottom row.
• Repeat the process of multiplying and adding across all columns.

The final row of numbers represents the coefficients of the quotient polynomial. The last number is the remainder. When dividing \( P(x) \) by \( x - c \), if the remainder is zero, then \( x - c \) is indeed a factor, confirming \( c \) as a root.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as the product of its factors, which are simpler, lower-degree polynomials. This process helps in solving polynomial equations and finding the roots. To factor a polynomial completely, one typically starts by factoring out the greatest common factor, if there is one, and then trying methods like trial and error, grouping, or using special formulas for quadratics and higher terms.
In the example, once synthetic division identifies \( x = 1 \) as a root, the polynomial can be divided into \( (x-1)(2x^2 - x - 3) \). The next step is to factor the remaining quadratic, \( 2x^2 - x - 3 \), by finding numbers that multiply to \( -6 \) and add to \( -1 \). You can then rewrite the middle term and apply factoring by grouping to break it into: \( (x + 1)(2x - 3) \). This thorough factorization provides useful insights into the polynomial’s structure and simplifies the process of finding its roots.

Roots of Polynomial

Finding the roots of a polynomial means identifying all the values for which the polynomial equals zero. These values, also known as the zeros or solutions, are important because they reveal where the graph of the polynomial intersects the x-axis.
The Rational Root Theorem helps to determine possible rational roots of a polynomial by considering the factors of the constant term and the leading coefficient. In our example, testing these potential roots through substitution revealed the roots \( x = 1, -1, \) and \( \frac{3}{2} \).
Each root corresponds to a factor of the polynomial. In this case, the complete factorization \( (x - 1)(x + 1)(2x - 3) \) directly pointed to these roots. The key to mastering polynomial roots is understanding both numerical and algebraic techniques to explore how polynomials behave and interact with their roots.