Problem 43 - College Algebra

Question

In Exercises \(39-52\), find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in $$ f(x)-x^{4}-2 x^{3}+x^{2}+12 x+8 $$

Step-by-Step Solution

Answer

The zeros of the given polynomial function are: \(x = -2\), \(x = 1\), and \(x = -1\pm \sqrt{5}\)

Step 1: Apply the Rational Zero Theorem

The rational zeros of a polynomial are potential zeros. The Rational Zero Theorem specifies a list of possible rational roots of the polynomial function \(f(x)=x^{4}-2 x^{3}+x^{2}+12 x+8\). According to this theorem, if a polynomial function has integer coefficients, then every rational zero will have the form \(p/q\) where \(p\) is a factor of the constant term (which is 8 here) and \(q\) is a factor of the leading coefficient (which is 1 here). So, the possible rational zeros are ±1, ±2, ±4, ±8.

Step 2: Test the Possible Zeros

The possible rational zeros are tested in the function to find the actual zeros. Here, plugging in \(x = -1, -2\), \(x = 1, 2\), and \(x = -4\), \(x = 4\) into \(f(x)\), you get \(f(-1) = 4\), \(f(-2) = 0\), \(f(1) = 12\), \(f(2) = 28\), \(f(-4) = 48\), and \(f(4) = 328\). So, \(x = -2\) is a zero of the polynomial.

Step 3: Polynomial Division

Once a zero is found, perform synthetic division or polynomial division to reduce the degree of the polynomial. After dividing the polynomial by \((x+2)\), you get \(x^{3} - 4x^{2} +9x +4\). Repeat Step 2 and Step 3 until you get a quadratic function.

Step 4: Solve the Quadratic Function

Since no other rational roots exist, set the resulting quadratic polynomial equal to zero and solve for \(x\). By factoring you get \((x-1)(x^2-4x-4)=0\). Hence the roots are \(x=1,-1\pm \sqrt{5}\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole-number exponents. Each term in a polynomial function is a coefficient multiplied by a variable with an exponent. The highest exponent in the polynomial is called the degree, which is crucial because it tells us about the function's behavior in terms of roots and graph shape.

• In the exercise given, our polynomial is of degree 4: \(x^4 - 2x^3 + x^2 + 12x + 8\).
• Higher degree polynomials can have more complex graphs and more roots compared to lower degree polynomials.

Learning how to find zeros of polynomial functions is key to understanding their behavior. We can identify zeros by using tools like the Rational Zero Theorem and synthetic division.

Descartes's Rule of Signs

Descartes's Rule of Signs is a handy mathematical rule used to predict the number of positive and negative real zeros in a polynomial function. By examining the sign changes in the polynomial's terms, we can estimate the possible number of real roots.

• Count the sign changes in the polynomial to estimate the number of positive real zeros.
• To count the possible negative real zeros, replace \(x\) with \(-x\) and count the sign changes again.

For our polynomial \(f(x) = x^4 - 2x^3 + x^2 + 12x + 8\), analyzing the sign changes can help narrow down where rational zeros are located, complementing the results from the Rational Zero Theorem.

Synthetic Division

Synthetic division is a simplified form of polynomial division that's easier and faster when dividing by a linear factor of the form \(x - k\). It helps confirm zeros and simplify the polynomial further to find more zeros.

• Write down the coefficients of the polynomial.
• Use a tested zero from the Rational Zero Theorem to perform the calculations.
• Bring down the first coefficient unchanged, multiply it by the zero, add to the next coefficient, and repeat.

In our problem, after identifying \(x = -2\) as a zero, synthetic division simplifies our polynomial to a lower degree \(x^3 - 4x^2 + 9x + 4\). This process is repeated until a quadratic is obtained.

Factoring Quadratics

Factoring quadratics is a method used to solve for zeros of a quadratic equation by expressing it as a product of two binomials. It's often the final step in finding polynomial zeros.

• Recognize the quadratic form \(ax^2 + bx + c\).
• Find two numbers that multiply to the constant term \(c\) and add up to the linear coefficient \(b\).
• Express the quadratic as \((x - p)(x - q) = 0\) and solve for roots \(x = p\) and \(x = q\).

In the exercise, the resulting quadratic after all other divisions is \(x^2 - 4x - 4\). Factoring it leads to solutions \(x - 1\) and the quadratic roots by completing the square or using the quadratic formula, resulting in \(x = -1 \pm \sqrt{5}\). Remember these techniques to solve any quadratic you encounter!