Problem 28 - Algebra for College Students

Question

Find a polynomial equation with real coefficients that has the given roots. $$-3, i \sqrt{3}$$

Step-by-Step Solution

Answer

The polynomial equation is ewlineLearn and simplify3) and learnKit)

Step 1: Recognize the root pairs

For polynomials with real coefficients, complex and imaginary roots come in conjugate pairs. Since one root is ewline3 is ewline3 is ewline, another root must be ewline3 is ewline.

Step 2: Write factors for each root

Each root corresponds to a factor of the polynomial. For the root -3, the factor is (x + 3). For the rootsewline3 ewlineLearnKitand its conjugateewline learnKit)

Step 3: Combine all factors into polynomial form

Multiply all the factors together to form the polynomial. Begin by multiplying the factors for the complex conjugate pair:ewlineewlineewlineewline)

Step 4 - Simplify the polynomial form

After combining and expanding the multiplication, we getewline learnKit ewline -1). Finally, multiply this result by the factor learned from step 3: learnKit)

Step 5 - Final polynomial equation

The polynomial equation that has the given roots ewline-3 andewlineLearn from step 4 = 01-0-0)

Key Concepts

Roots of Polynomials

Understanding the **roots of polynomials** is crucial when solving polynomial equations. If you have a polynomial equation, the roots are the values of \(x\) that make the equation equal to zero.
For example, if \(x = -3\) and \(x = i\sqrt{3}\) are the roots of a polynomial, it means that plugging these values into the polynomial equation will result in zero. This is fundamental in algebra and helps solve many problems involving polynomials.
In a polynomial with real coefficients, if one of the roots is a complex number, its *complex conjugate* must also be a root. This leads us to our next concept.

Complex Conjugate Roots

Polynomials with real coefficients require that non-real roots occur in **complex conjugate pairs**.
A complex conjugate of a number \(a + bi\) is \(a - bi\). For instance, if you have \(i\sqrt{3}\) as a root, then \(-i\sqrt{3}\) must also be a root.
This concept ensures that the polynomial remains having real coefficients, which is often a requirement in many mathematical problems and applications.
So for our example, given \(i\sqrt{3}\), we automatically know that \(-i\sqrt{3}\) is also a root.

Factors of Polynomials

Finding the **factors of polynomials** is a necessary step to form the complete polynomial equation.
Every root \(x = r\) gives a factor of \((x - r)\). For example:

• The root \(-3\) gives the factor \((x + 3)\)
• The root \(i\sqrt{3}\) gives the factor \((x - i\sqrt{3})\)
• The root \(-i\sqrt{3}\) gives the factor \((x + i\sqrt{3})\)

Combining these factors, you multiply them together to get the polynomial:
\[ (x + 3)(x - i\sqrt{3})(x + i\sqrt{3}) \]
By multiplying the complex conjugates first, you simplify the expression:
\[ (x - i\sqrt{3})(x + i\sqrt{3}) = x^2 + 3 \]
Finally, multiply by the remaining factor:
\[ (x + 3)(x^2 + 3) \]
The result is the polynomial equation in its simplest form:
\[ P(x) = x^3 + 3x^2 + 9x + 9 \]