6.12: Finding Imaginary Solutions
Louis calculates that the area of a rectangle is represented by the equation
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James Sousa: Ex 4: Find the Zeros of a Polynomial Function with Imaginary Zeros
Guidance
In #12 from the previous problem set, there are two imaginary solutions. Imaginary solutions always come in pairs. To find the imaginary solutions to a function, use the Quadratic Formula. If you need a little review on imaginary numbers and how to solve a quadratic equation with complex solutions see the Quadratic Equations chapter.
Example A
Solve
Solution: First, this quartic function can be factored just like a quadratic equation. See the Factoring Polynomials in Quadratic Form concept from this chapter for review.
Now, because neither factor can be factored further and there is no
Including the imaginary solutions, there are four, which is what we would expect because the degree of this function is four.
Example B
Find all the solutions of the function
Solution: When graphed, this function does not touch the
Now, set each factor equal to zero and solve.
Example C
Find the function that has the solution 3, 2, and
Solution: Notice that one of the given solutions is imaginary. Imaginary solutions always come in pairs, so
Any multiple of this function would also have these roots. For example,
Intro Problem Revisit First we need to change the equation to standard form. Then we can factor it.
Solving for x we get
All of the solutions are imaginary and the area of a rectangle must have real solutions. Therefore Louis did not calculate correctly.
Guided Practice
Find all the solutions to the following functions.
1.
2.
3. Find the equation of a function with roots 4,
Answers
1. First, graph the function.
Using the Rational Root Theorem, the possible realistic zeros could be
Of these three possibilities, only 4 is a zero. The leftover polynomial,
2.
Setting each factor equal to zero, we have:
3. Recall that irrational and imaginary roots come in pairs. Therefore, all the roots are 4,
Practice
Find all solutions to the following functions. Use any method.

f(x)=x4+x3−12x2−10x+20 
f(x)=4x3−20x2−3x+15 
f(x)=2x4−7x2−30 
f(x)=x3+5x2+12x+18 
f(x)=4x4+4x3−22x2−8x+40 
f(x)=3x4+4x2−15 
f(x)=2x3−6x2+9x−27 
f(x)=6x4−7x3−280x2−419x+280 
f(x)=9x4+6x3−28x2+2x+11 
f(x)=2x5−19x4+30x3+97x2−20x+150
Find a function with the following roots.

4,i 
−3,−2i 
5√,−1+i 
2,13,4−2√  Writing Write down the steps you use to find all the zeros of a polynomial function.
 Writing Why do imaginary and irrational roots always come in pairs?

Challenge Find all the solutions to
f(x)=x5+x3+8x2+8 .
Complex Conjugate
Complex conjugates are pairs of complex binomials. The complex conjugate of is . When complex conjugates are multiplied, the result is a single real number.complex number
A complex number is the sum of a real number and an imaginary number, written in the form .conjugate pairs theorem
The conjugate pairs theorem states that if is a polynomial of degree , with and with real coefficients, and if , where , then . Where is the complex conjugate of .fundamental theorem of algebra
The fundamental theorem of algebra states that if is a polynomial of degree , then has at least one zero in the complex number domain. In other words, there is at least one complex number such that . The theorem can also be stated as follows: an degree polynomial with real or complex coefficients has, with multiplicity, exactly complex roots.Imaginary Number
An imaginary number is a number that can be written as the product of a real number and .Imaginary Numbers
An imaginary number is a number that can be written as the product of a real number and .Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Roots
The roots of a function are the values of x that make y equal to zero.Zero
The zeros of a function are the values of that cause to be equal to zero.Zeroes
The zeroes of a function are the values of that cause to be equal to zero.Image Attributions
Here you'll find all the solutions to any polynomial, including imaginary solutions.
Concept Nodes:
Complex Conjugate
Complex conjugates are pairs of complex binomials. The complex conjugate of is . When complex conjugates are multiplied, the result is a single real number.complex number
A complex number is the sum of a real number and an imaginary number, written in the form .conjugate pairs theorem
The conjugate pairs theorem states that if is a polynomial of degree , with and with real coefficients, and if , where , then . Where is the complex conjugate of .fundamental theorem of algebra
The fundamental theorem of algebra states that if is a polynomial of degree , then has at least one zero in the complex number domain. In other words, there is at least one complex number such that . The theorem can also be stated as follows: an degree polynomial with real or complex coefficients has, with multiplicity, exactly complex roots.Imaginary Number
An imaginary number is a number that can be written as the product of a real number and .Imaginary Numbers
An imaginary number is a number that can be written as the product of a real number and .Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Roots
The roots of a function are the values of x that make y equal to zero.Zero
The zeros of a function are the values of that cause to be equal to zero.Zeroes
The zeroes of a function are the values of that cause to be equal to zero.