High School: Number and Quantity

The Complex Number System HSN-CN.C.9

9. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Once upon a time, students were given easy equations to solve, equations with one variable to the first power. They found one answer, checked it to make sure it was correct, and all was right with the world.

Then came the "F" word. Factoring. And along with it came quadratic equations and their two answers. Then they got into cubics, with three answers and… wait. Do we sense a pattern, here?

Well, the Fundamental Theorem of Algebra certainly thinks so. It says that the degree of a polynomial function is the same as the number of roots. So a linear equation (one with x to the first power) has 1 root. Quadratics (with x to the second power) have 2 roots. Cubics (with x to the third power) have 3 roots. And so it goes.

Regardless of how high the degree, the degree matches the number of roots.

But… (yeah, you knew there had to be a "but") it doesn't say they have to be different roots, necessarily. For example, if our equation is x2 + 4x + 4 = 0, we're going to get (x + 2)(x + 2) = 0, and our roots will both be -2. Sure there are 2 of them, but they're identical.

The Fundamental Theorem of Algebra also doesn't say that the roots have to be real, so make sure your students know how to work in the realm of the imaginary.

The one good thing about complex roots (and irrational roots, too): they come in conjugate pairs. So if x = 7i is one root, we automatically know that x = -7i is another. So if we have an odd number of roots, at least one of them has to be rational.

Drills

1. How many roots does the equation 8x7 – 6x5 + 4x3 + 2x + 1 = 0 have?

7

The number of roots is defined by the degree (the highest exponent). Since the degree here is 7, the equation has 7 roots.

2. In the same equation, 8x7 – 6x5 + 4x3 + 2x + 1 = 0, what's the maximum number of complex roots?

6

We know that there are only 7 roots total, so it's impossible to have more than that. Complex roots come in conjugate pairs, so there can only be an even number of them. That means the maximum number of complex roots would be (B).

3. How many roots does the equation 4x + 3 = x3 have?

3

Just because the variables are on opposite sides of the equal sign doesn't change anything. We still have to look at the highest power for the number of roots. That means there will be a maximum of 3 roots for this equation.

4. What is the maximum number of complex roots that 4x + 3 = x3 can have?

2

The maximum number of complex roots possible in any polynomial is the largest even number that is equal to or less than the highest degree. That's because complex numbers always come in conjugates, so there has to be an even number of them. Since the degree of the polynomial is 3, there can only be up to 2 complex roots.

5. What is the maximum number of complex roots that 7x12 + 3x5 = 19x14 – 3 can have?

14

The highest exponent this polynomial contains is 14. That means there exist 14 roots for this polynomial. Of these roots, there can only be an even number of complex roots. Since 14 is already an even number, it is possible for this polynomial to have only imaginary roots.

6. How many real roots will 10x19 + 3 = 2x11 + 15x16 – 4x3 have if it has the maximum number of complex roots possible?

1

If a polynomial has the maximum number of imaginary roots, the number of real roots depends on the degree of the polynomial (and no, we don't mean whether it got a Bachelor's or a Master's). Since the highest exponent in the polynomial is 19 and the number of complex roots must be even, that leaves 1 real root left over.

7. Why would factoring the sum of perfect squares guarantee that the imaginary roots come in conjugate pairs?

Because factoring the difference of two squares gives conjugate pairs

Factoring the difference of perfect squares (as in x2a2) always yields (xa)(x + a). So you'll always end up with conjugate pairs. Since we're factoring the sum of perfect squares, we'll end up with an imaginary number for a.

8. Why does the quadratic formula always give two answers for complex roots?

Because of the ± before the discriminant

When we use the quadratic formula, we always get . When the portion under the radical (called the discriminant) is negative, that means our roots are imaginary. The ± before it ensures we'll have two conjugates as our answer.

9. Is it possible for a quadratic equation to have one real root and one complex root?

No, because all equations must have an even number of complex roots

Since complex numbers come in conjugates, there must always be an even number of them. Think of them as the animals on Noah's ark: they come in pairs. Of course, that means it's impossible for us to have just one. Quadratic equations can definitely have them, and there can be some real and some imaginary. So the answer is (D).

10. How would it be possible to get identical roots?