Problem 16 - A Graphical Approach to Precalculus with Limits

Question

One or more zeros are given for each polynomial. Find all remaining zeros. \(P(x)=2 x^{4}-x^{3}-27 x^{2}+16 x-80 ;-4\) and 4 are Zeros.

Step-by-Step Solution

Answer

The zeros are \(-4, 4, \sqrt{5}, -\sqrt{5}\).

Step 1: Understand Given Zeros

We are given that the polynomial \(P(x) = 2x^4 - x^3 - 27x^2 + 16x - 80\) has zeros at \(-4\) and \(4\). This means \(x+4\) and \(x-4\) are factors of \(P(x)\).

Step 2: Use Factoring by Synthetic Division

First, use synthetic division to divide \(P(x)\) by \(x-4\), one given factor. The remainder should be zero if \(4\) is truly a zero. Perform the division and verify that it divides evenly.

Step 3: Divide by Second Zero

Next, use synthetic division to divide the quotient obtained from Step 2 by \(x+4\). Again, verify that the remainder is zero to ensure \(-4\) is also a zero.

Step 4: Factor the Quadratic

After two synthetic divisions, a quadratic polynomial remains. Factor this quadratic using either factoring methods or the quadratic formula to find the remaining zeros.

Step 5: Verify All Zeros

Combine all the zeros you have found. Verify by plugging these zeros back into the original polynomial equation to ensure they all satisfy \(P(x) = 0\).

Key Concepts

Synthetic Division

Synthetic division is a powerful tool used to divide polynomials, especially when you have a known zero. It simplifies the process compared to traditional long division. In the given exercise, we know that -4 and 4 are zeros of the polynomial. Therefore, we can use synthetic division to divide the polynomial by these linear factors,

• When you divide by a factor like \( x - 4 \), you set the zero, \( 4 \), to the left of the division. For each term of the polynomial's coefficients, bring down the first coefficient, multiply it by the zero, add it to the next coefficient, and repeat.
• After going through all coefficients, the last number is the remainder. If your division is correct and the zero is indeed a factor, the remainder should be zero.
• Repeat the process with the second given zero, \( -4 \).

This method is efficient, particularly for higher-degree polynomials, allowing you to quickly reduce them to a simpler polynomial.

Factoring Polynomial

Factoring polynomials involves rewriting them as a product of simpler polynomials. This is critical in finding the zeros of a polynomial function. After using synthetic division twice, the given polynomial is reduced, making it easier to factor further.

• Once you have a quadratic polynomial, you can attempt to factor it into two binomials. Look for two numbers that multiply to give the constant term and add to give the middle coefficient.
• If the quadratic does not factor easily with integers, you might use other methods, such as completing the square or the quadratic formula, to find its roots.
• Recognizing patterns such as the difference of squares or perfect square trinomials can also help in faster factoring.

Factoring is not just a skill but also an art and becomes easier with practice by recognizing these patterns.

Quadratic Formula

The quadratic formula is a tried-and-true method to find the roots of any quadratic equation, especially when factoring is complicated or impossible.

• The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
• Use the quadratic formula when synthetic division leaves you with a quadratic that does not factor easily. It will provide the solutions directly, as long as the discriminant \( b^2 - 4ac \) is non-negative.
• If the discriminant is zero, there is one real root. If positive, there are two distinct real roots. A negative discriminant implies complex roots.

This is an essential tool for solving quadratic equations, as it always provides a solution, offering great reliability when dealing with polynomial zeros.

Polynomial Roots

The roots, or zeros, of a polynomial are the values of \( x \) that make \( P(x) = 0 \). Finding these indicates locations where the polynomial intersects the x-axis on a graph.

• These roots can be real or complex. In this exercise, by using methods like synthetic division, factoring, and the quadratic formula, we find real roots.
• Verify found roots by substituting back into the polynomial and checking if the result is zero. All zeros should satisfy \( P(x) = 0 \).
• Finally, recognize that knowing all polynomial roots helps in graphing and understanding the polynomial's behavior, such as its shape and the number of x-axis crossings.

Understanding polynomial roots is a vital part of algebra, as it allows one to interpret and solve polynomial equations comprehensively, providing deep insight into their mathematical structure.