Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the given zero to find all the zeros of the function. Function $$f(x)=25 x^{3}-55 x^{2}-54 x-18$$ Zero $$\frac{1}{5}(-2+\sqrt{2} i)$$

Step-by-Step Solution

Answer

The roots of the function \(f(x) = 25x^3 - 55x^2 - 54x - 18\) are \(\frac{1}{5}(-2+\sqrt{2} i)\), \(\frac{1}{5}(-2-\sqrt{2} i)\), and \(2\).

Step 1: Identify the Conjugate

Identify the conjugate of the given zero. Valid complex zeros always come in conjugate pairs due to the Complex Conjugate Root Theorem. So, if \(\frac{1}{5}(-2+\sqrt{2} i)\) is a zero of the polynomial, its conjugate \(\frac{1}{5}(-2-\sqrt{2} i)\) is also a zero.

Step 2: Formulate Quadratic Factor

Generate a quadratic factor from the given complex zero and its conjugate. This gives the factor \((x - \frac{1}{5}(-2+\sqrt{2} i))(x - \frac{1}{5}(-2-\sqrt{2} i))\). After simplification, this yields the factor \(25x^2 - 10x + 1\).

Step 3: Divide the Polynomial

Use the quadratic factor to divide the polynomial \(f(x)\) with synthetic division. The divisor is \(25x^2 - 10x + 1\) and the dividend is \(25x^3 - 55x^2 - 54x - 18\). This yields a quotient of \(x - 2\).

Step 4: Identify the Remaining Zero

Set the quotient equal to zero to solve for the remaining root. For \(x - 2 = 0\), the solution is \(x = 2\).

Key Concepts

Synthetic Division

Synthetic division is a streamlined method for dividing polynomials when you want to divide a polynomial by a linear divisor. In our context, it's particularly helpful after you've identified some zeros of the polynomial, like complex conjugates, and want to simplify your polynomial before solving for other roots. Here's how it works:

• Write down the coefficients of the polynomial you wish to divide.
• The number you're dividing by (the zero you've already found) is placed to the left side.
• Perform the synthetic division calculation, which involves successive steps of multiplication and addition of these coefficients.

This method helps reduce calculation errors and is much faster than traditional long division with polynomials. In our example, synthetic division helps us separate the quadratic factor from the cubic polynomial and simplifies the problem of finding the remaining roots.

Quadratic Factors

Quadratic factors arise when you have found a complex conjugate pair as zeros of a polynomial. Since complex roots in polynomials come in conjugate pairs, these can be neatly captured in a quadratic expression. This quadratic factor is created by multiplying the expressions \((x - rac{1}{5}(-2+\sqrt{2}i))(x - rac{1}{5}(-2-\sqrt{2}i))\),which is the product of the stated conjugate pair. Upon simplification, this yields a quadratic factor \(25x^2 - 10x + 1\).Quadratic factors are crucial because they help break down the polynomial into simpler pieces, making it easier to find all the polynomial roots. They particularly help when dealing with imaginary roots, converting them into real-number coefficients.

Polynomial Roots

Finding the roots of a polynomial, which are simply the solutions for which the polynomial equals zero, can be made more manageable through strategic use of discovered zeros and synthetic division. To identify the roots:

• Start with the known zeros, like our complex conjugate pair.
• Create a quadratic factor from these zeros.
• Use this factor to perform synthetic division on the polynomial to find the remainder and quotient.
• Set the quotient to zero and solve for any remaining roots.

In this example, after identifying our complex zeros and using synthetic division, we found an additional real root \(x = 2\).Understanding these steps allows you to precisely locate each root, ensuring no solutions are overlooked.