Problem 29 - College Algebra

Question

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 obtaining the first zero or the first root. $$ f(x)=x^{3}-4 x^{2}-7 x+10 $$

Step-by-Step Solution

Answer

The zeros of the polynomial \(f(x)=x^{3}-4 x^{2}-7 x+10\) are -2, 1, and 5.

Step 1: Use Rational Zero Theorem

According to the Rational Zero Theorem, the possible rational zeros of a polynomial are all the ratios between the factors of the constant term (10) and the factors of the highest order term's coefficient (1). The factors of 10 are ±1, ±2, ±5, ±10. Since the leading coefficient is 1, all these numbers are possible rational zeros of the polynomial.

Step 2: Use Descartes's Rule of Signs

To find the number of possible positive and negative zeros, apply Descartes's Rule of Signs. Counting the number of sign changes in \(f(x)\) will give us the possible number of positive real roots, and counting the sign changes in \(f(-x)\) will give us the possible number of negative real roots. Plug in -x into the equation and simplify to obtain \(f(-x)=-x^{3}-4 x^{2}+7 x+10\). \(f(x)\) exhibits two sign changes, which means there are either two or zero positive roots. \(f(-x)\) exhibits one sign change, which means there is exactly one negative root.

Step 3: Test the possible rational zeros

Plug each possible rational zero from Step 1 back into the original equation and find which yield 0. The first rational zero that works is -2.

Step 4: Use Synthetic Division

By using synthetic division, divide the polynomial by the factor associated with the root found in step 3, which is \(x + 2\). The result will be a quadratic equation.

Step 5: Find Remaining Roots

Solve the quadratic equation from Step 4 to find the other roots. In this case, the quadratic equation from synthetic division is \(x^2 - 6x + 5\). It factors into \((x-1)(x-5)\), which means the other roots of the original polynomial are x=1 and x=5.

Key Concepts

Rational Zero Theorem

When you're tasked with finding all the zeros of a polynomial like \( f(x) = x^3 - 4x^2 - 7x + 10 \), the Rational Zero Theorem can be an extremely helpful starting point. This theorem gives us a way to list all possible rational zeros of a polynomial function based on its coefficients.

Simply put, you can find these potential zeros by looking at the ratio of factors of the constant term (the term without any variable, which is 10 in our example) to the factors of the leading coefficient (the coefficient of the term with the highest exponent, which is 1 for \( x^3 \)). This means for our polynomial, the possible rational zeros are ±1, ±2, ±5, and ±10. These candidates are the stepping stones for finding the actual zeros of the polynomial.

Descartes's Rule of Signs

The next strategic step is to utilize Descartes's Rule of Signs, a principle that narrows down the number of possible positive and negative real zeros in a polynomial. The rule requires you to count the number of times the signs of the terms change.

For the positive zeros, look at the original polynomial \( f(x) \). Our polynomial, \( f(x) = x^3 - 4x^2 - 7x + 10 \), changes signs from positive to negative and then back to positive, resulting in two sign changes. This suggests there could be two or zero positive real zeros.

To find the possible negative zeros, substitute \( x \) with \( -x \) to get \( f(-x) \) and count the sign changes there. Doing this for our polynomial gives us one sign change, indicating one negative real zero. Understanding the output from Descartes's Rule of Signs helps narrow down the search and speeds up the zero finding process.

Synthetic Division

After identifying a possible zero using the Rational Zero Theorem and considering Descartes's Rule of Signs, you'll want to confirm it. This is where Synthetic Division comes in, a shorthand method of dividing polynomials that's quicker and simpler than the long division you may be used to.

Using the identified zero, you'll test it by dividing the polynomial function to see if the remainder is zero, confirming it as a true zero of the function. In our exercise, when -2 was tested, it proved to be a zero, so we then used synthetic division to divide our polynomial by \( x + 2 \), the corresponding factor.

This process turns our third-degree polynomial into a more manageable quadratic one, which we can then solve to find the remaining roots. For the given polynomial, synthetic division reduces it to \( x^2 - 6x + 5 \), which further factors down to \( (x-1)(x-5) \), revealing the other zeros, x=1 and x=5. Mastering synthetic division is a crucial step in efficiently solving polynomial equations.