Problem 4 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=2 x^{3}-7 x^{2}-17 x+10 $$

Step-by-Step Solution

Answer

The only rational zero is \( x = -1 \).

Step 1: Identify Rational Root Theorem Candidates

The Rational Root Theorem states that any rational solution \( \frac{p}{q} \) of the polynomial equation is such that \( p \) divides the constant term \( a_0 = 10 \) and \( q \) divides the leading coefficient \( a_n = 2 \). Thus, the possible values for \( p \) are the factors of 10: \( \pm 1, \pm 2, \pm 5, \pm 10 \). The possible values for \( q \) are the factors of 2: \( \pm 1, \pm 2 \). Hence, possible rational zeros are \( \pm 1, \pm \frac{1}{2}, \pm 2, \pm 5, \pm \frac{5}{2}, \pm 10 \).

Step 2: Test Candidates Using Synthetic Division

To check which candidates are zeros, use synthetic division on each candidate. Start with \( x = 1 \). By synthetic division, the polynomial evaluated at 1 is not zero. Check for \( x = -1, x = \frac{1}{2}, x = -\frac{1}{2}, x = 2 \), etc., until you find a zero.

Step 3: Confirm Zero (-1) by Synthetic Division

Using synthetic division for \( x = -1 \), we find that the remainder is 0: \[\begin{array}{r|rrrr}-1 & 2 & -7 & -17 & 10\ & & -2 & 9 & 8 \\hline & 2 & -9 & -8 & 0 \\end{array}\]Since the remainder is 0, \( x = -1 \) is a zero of \( f(x) \).

Step 4: Factor the Polynomial Using the Zero

Since \( x = -1 \) is a factor, we can write the reduced polynomial from step 3 as \( 2x^2 - 9x - 8 \). Now, we need to find the zeros of this quadratic polynomial.

Step 5: Solve the Quadratic Equation

The quadratic equation \( 2x^2 - 9x - 8 = 0 \) can be solved using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] With \( a = 2, b = -9, c = -8 \), the solutions are:\[ x = \frac{9 \pm \sqrt{(9)^2 - 4(2)(-8)}}{4} \]\[ x = \frac{9 \pm \sqrt{113}}{4} \]These are not rational numbers.

Step 6: List All Rational Zeros

Since the only rational solution found was \( x = -1 \) and the quadratic roots are irrational, the only rational zero of the polynomial \( f(x) = 2x^3 - 7x^2 - 17x + 10 \) is \( x = -1 \).

Key Concepts

Polynomial function

A polynomial function is an algebraic expression consisting of variables, coefficients, and exponents seamlessly combined. Each variable in a polynomial has a non-negative integer exponent and appears in a term. The general form of a polynomial function is: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] where:

• \( a_n, a_{n-1}, ..., a_0 \) are coefficients, which can include rational numbers.
• \( x \) is the variable.
• \( n \) represents the degree of the polynomial, which is determined by the highest exponent.

In our exercise, the polynomial function is \( f(x) = 2x^3 - 7x^2 - 17x + 10 \). It is a cubic polynomial, since the highest power is 3. Understanding polynomial functions forms a crucial basis for solving equations and analyzing functions.

Synthetic division

Synthetic division is a simplified method of dividing a polynomial by a divisor of the form \( x - r \), where \( r \) is a root of the polynomial. This technique is quicker and easier than traditional long division. To carry out synthetic division:

• Write the coefficients of the dividend polynomial in descending order of exponents.
• Select the value of \( r \) from the Rational Root Theorem candidates.
• Bring down the leading coefficient to the bottom row.
• Multiply \( r \) by the number just written in the bottom row, place the result under the next coefficient, and then add these numbers together.

Continue the process for each coefficient. If the final remainder is 0, the candidate is a root. In our case, synthetic division confirmed \( x = -1 \) as a root, transforming the polynomial to a quadratic form \( 2x^2 - 9x - 8 \). This method ensures efficient root-finding while simplifying the polynomial.

Quadratic formula

The quadratic formula is a crucial tool for finding the roots of any quadratic equation in the form \( ax^2 + bx + c = 0 \). The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where:

• \( a \) is the coefficient of \( x^2 \).
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.
• The "\( \pm \)" symbolizes the two potential solutions.

In our polynomial, after confirming \( x = -1 \) as a zero, we used the quadratic formula to solve for roots of \( 2x^2 - 9x - 8 = 0 \). This culminated in solutions involving \( \sqrt{113} \), thus indicating they are irrational. This method enables precise calculation of roots, even when manual factoring isn't feasible.

Rational zeros

Understanding rational zeros is key to solving polynomial equations. A rational zero of a polynomial is any zero that can be expressed as a fraction \( \frac{p}{q} \), where both \( p \) and \( q \) are integers, with \( q eq 0 \). The Rational Root Theorem provides a method to find possible rational zeros. By examining factor combinations of the polynomial’s leading coefficient and constant term:

• Use the factors of the constant term (the last term of the polynomial).
• Use the factors of the leading coefficient (the first term with the highest power).
• Generate potential rational roots as \( \frac{\text{factor of constant}}{\text{factor of leading coefficient}} \).

For our polynomial \( f(x) = 2x^3 - 7x^2 - 17x + 10 \), only \( x = -1 \) was confirmed as a rational zero. Understanding these potential roots can dramatically streamline solving polynomials.