Sketching Polynomials Examples

Sketch a rough graph of y = -x⁴ – 4x³ + 14x² + 36x – 45.

We'll start off with Descartes' Rule of Signs. We have 2 sign changes in the original polynomial, so we have 2 or 0 positive real roots.

f(-x) = -(-x)⁴ – 4(-x)³ + 14(-x)² + 36(-x) – 45

= -x⁴ + 4x³ + 14x² – 36x – 45

Again, we have 2 sign changes here, so we have either 2 or 0 negative real roots. The next step is the Rational Root Theorem, and it says that our possible rational roots are:

±1, ±3, ±5, ±9, ±15, ±45

Time for the dirty part of all this, digging for roots. Using our wonder shovel, synthetic division, on x = 1:

We found one, and it's 1. Our quotient is -x³ – 5x² + 9x + 45. How about some more synthetic division? Let's try x = -1.

That's no root, it's a space station. No, wait, it isn't that either. It also isn't a lower bound; the signs in the quotient do not alternate. Let's go for another point, x = 3.

Second root, unlocked. We are now down to a quadratic equation, -x² – 8x – 15. We can factor this to

-(x + 5)(x + 3)

Now we can pull our entire polynomial into one big, fully factored mess:

y = -(x – 1)(x – 3)(x + 3)(x + 5)

We have found all four roots for our 4-degree polynomial. This is an even polynomial with a negative coefficient. We will wave our arms to floor like we just don't care. The multiplicity for every zero is 1, so they all cross over to the other side.

Sketch a rough graph of y = (2x – 1)²(x + 1)⁴(x² + 2).

We have the roots already given to us. Nice. The first and third factors need to be solved for x, but we don't mind. The first is x = \frac{1}{2}, and it has a multiplicity of 2. We also have:

Ooh, a pair of complex roots. They don't cross the x-axis, but they do count towards our total number of roots.

The last root is x = -1, with a multiplicity of 4.

At this point, we can start sketching. The graph's end behavior is: it has even degree and a positive leading coefficient. Both arms go in the air. But we do care; otherwise we wouldn't have noticed.

This is a polynomial with a degree of 9. What are the real zeros of the polynomial, and what can we tell about their multiplicity? Is the polynomial even or odd?

We have to work backwards this time, from the graph to learning something about the polynomial's equation.

The polynomial is odd, and it has a positive leading coefficient. We can tell from the pixels, and from having seen a number of polynomials in our day. Also, the left arm is pointing down and the right arm is pointing up. That too.

The zeros are at x = -2, with some even multiplicity, and x = 4, with some odd multiplicity. The multiplicity for x = 4 is 3; it would be flatter otherwise. We can't say for sure what the multiplicity is for -2, though, just that it is larger than 2. Some of the roots might be complex, and we wouldn't be able to see those on the graph. So, x = -2 has a multiplicity of either 4 or 6. We can only go up to 9 total.