Problem 63 - A Graphical Approach to College Algebra

Question

Find all rational zeros of each polynomial function. $$P(x)=x^{3}+\frac{1}{2} x^{2}-\frac{11}{2} x-5$$

Step-by-Step Solution

Answer

The rational zero is \(-5\).

Step 1: Determine Possible Rational Zeros

According to the Rational Root Theorem, any rational zero of the polynomial \(P(x) = x^3 + \frac{1}{2}x^2 - \frac{11}{2}x - 5\) is of the form \(\frac{p}{q}\) where \(p\) is a factor of the constant term (-5) and \(q\) is a factor of the leading coefficient (1). Therefore, the possible rational zeros are \(\pm 1\), \(\pm 5\).

Step 2: Evaluate Possible Rational Zeros

Substitute each possible rational zero into the polynomial to see if it equals zero:- For \(x = 1\):\[P(1) = 1^3 + \frac{1}{2}(1)^2 - \frac{11}{2}(1) - 5 = 1 + \frac{1}{2} - \frac{11}{2} - 5 = -9\]- For \(x = -1\):\[P(-1) = (-1)^3 + \frac{1}{2}(-1)^2 - \frac{11}{2}(-1) - 5 = -1 + \frac{1}{2} + \frac{11}{2} - 5 = 1\]- For \(x = 5\):\[P(5) = 5^3 + \frac{1}{2}(5)^2 - \frac{11}{2}(5) - 5 = 125 + \frac{25}{2} - \frac{55}{2} - 5 = 45\]- For \(x = -5\):\[P(-5) = (-5)^3 + \frac{1}{2}(-5)^2 - \frac{11}{2}(-5) - 5 = -125 + \frac{25}{2} + \frac{55}{2} - 5 = 0\]\(x = -5\) is a zero.

Step 3: Verify the Zero with Polynomial Division

Perform synthetic division of \(P(x)\) by \(x + 5\) to confirm \(-5\) as a zero and to find other zeros.Using synthetic division:- Coefficients: \(1, \frac{1}{2}, -\frac{11}{2}, -5\)- Synthetic division steps: 1. Bring down 1. 2. Multiply \(-5\) with 1 and add to \(\frac{1}{2}\): New term \(-5 + \frac{1}{2} = -\frac{9}{2}\). 3. Multiply \(-5\) with \(-\frac{9}{2}\) and add to \(-\frac{11}{2}\): New term \(\frac{45}{2} - \frac{11}{2} = 17\). 4. Multiply \(-5\) with 17 and add to \(-5\): New term \(-85 - 5 = 0\).The quotient polynomial is \(x^2 - \frac{9}{2}x + 17\), which gives no more rational zeros.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool for finding rational zeros of a polynomial function. It tells us that any rational solution of a polynomial equation is a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
This theorem helps narrow down the search for potential zeros, making the problem more manageable.
In our example, the constant term of \( P(x) = x^3 + \frac{1}{2}x^2 - \frac{11}{2}x - 5 \) is \(-5\), and the leading coefficient is \(1\). This means the possible rational zeros are \( \pm 1 \) and \( \pm 5 \).
By using the Rational Root Theorem, we reduce the complexity of guessing which numbers could be zeros.

Synthetic Division

Synthetic division is a shortcut method for dividing polynomials, particularly when you're working with possible rational roots.
It's simpler and quicker than traditional long division.
This method involves writing down the coefficients of the polynomial and using the potential zero to simplify the division process.
For instance, to confirm \( x = -5 \) as a zero of the polynomial \( P(x) \), we perform synthetic division with \(-5\).
Through organized steps—such as bringing down coefficients, multiplying, and adding—we verify whether the division results in a remainder of zero. If it does, \(-5\) is a genuine zero, and the quotient polynomial helps find any other zeros.

Polynomial Function

A polynomial function is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
In our example, the polynomial function \( P(x) = x^3 + \frac{1}{2}x^2 - \frac{11}{2}x - 5 \) is a cubic polynomial, meaning its highest exponent is 3.
Such functions can have up to three zeros, which can be real or complex.
Understanding the structure of polynomial functions helps in analyzing their roots, behavior, and graph characteristics.
This also involves recognizing the degree and leading coefficient, which play critical roles in calculating rational zeros.

Factors

Factors of a polynomial are expressions that multiply together to result in the original polynomial.
Finding the zeros of a polynomial involves breaking it into smaller expressions or factors.
If a polynomial has a zero, \( x - c \), it means \( x = c \) is a factor.
Using synthetic division to verify \(-5\) as a zero shows that \( x + 5 \) is a factor of \( P(x) \).
The quotient \( x^2 - \frac{9}{2}x + 17 \) from synthetic division didn't provide any additional rational factors, meaning further exploration might involve complex or irrational roots depending on further factorization.