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### Course: Algebra (all content)>Unit 10

Lesson 27: Advanced polynomial factorization methods

# Factoring higher-degree polynomials

Sal factors p(x)=2x^5+x^4-2x-1 as (2x+1)(x^4-1) using grouping. Then he further factors (x^4-1) as (x^2+1)(x+1)(x-1) using the special product form of difference of squares. Created by Sal Khan.

## Want to join the conversation?

• At , it says "x^2=-1." That means that x = imaginary number i. How could we graph i on a coordinate plane?
• When graphing complex numbers, there is an axis for the real portion and another axis for the imaginary portion. If I remember correctly, the x-axis will be your real axis and the y-axis will be the imaginary axis. For example, if I have the complex number 1 + 2i, the point will look like the point (1, 2) on a set of real axes.
• If a real zero is a place where the function intersects the x-axis then does the function sal solved intersect the x axis 3 times?
• yes it does because the equation is a 5th degree polynomial so it can intersect it at maximum of 5 times.
• The Fundamental Theorem of Algebra tells that any polynomial of degree n has exactly n roots, but Sal in the video found 3 roots for the polynomial of 5 degree. Please explain how is this possible?
• As I'm sure you know, the Fundamental Theorem of Algebra doesn't say that all the roots have to be real.
In this case there are two complex roots corresponding to the (x² + 1) factor, namely ±i
• Isn't there a set a rules one could follow when factoring x^n power? It seems that the way in which you factor x^2 is completely distinct from how you would factor x^3, or x^4 and so on. There seems to be no general order to factoring exponents, which is what I really do not gel with...
• Unfortunately, there is NOT a set a rules for factoring when you get to higher powers. Grouping can sometimes work--and when it does, it works well--but that method requires a pattern in the numbers. What mathematicians do to find roots in the more difficult cases is to a) use synthetic division with the rational root theorem to guess and check possibilities for rational roots or b) use graphing to identify the real roots (which might only lead to an approximation, not an exact root).

There are (much more difficult) formulas like the quadratic formula for degree x^3 and x^4, but it's actually a deep mathematical theorem (and fascinating historical story) that there can be no formula for degree x^5 polynomials or higher. So unfortunately, even though the Fundamental Theorem guarantees roots, practically speaking, it might be impossible to identify them sometimes.
• when he further factored (x^4-1) to (x^2-1) and (x^2+1) why is that necessary?
• Karma's correct. (See other reply)In order to completely simplify (and receive full marks on tests), you must be able to identify difference of squares, sum of squares, and whatnot. My advice is to memorize the form of such things to be able to find them in problems. Also, if you can factor anything out, do so right away to get one step closer to the final answer.
• h(x)=-x^3-5x^2 how do I find the zeros of the function
• How do I find zeros? and factoring them out
(1 vote)
• If you get an imaginary root, how would you graph that on the coordinate system?
• You can't. Imaginary numbers can only be graphed on a complex plane. Typically you would see something like i replacing the y-direction. You could draw another coordinate system to suit your needs or even draw a z-direction and plot it in 3 dimensions. Then you could plot i and -i on the new imaginary direction.
• How to factor x^4 + 2x^3 + 2x^2 + 2x + 1
• Guess-and-checking a few simple numbers, I found that i is a root. Because this polynomial has real coefficients, that means that the complex conjugate -i is also a root. So we can factor out
(x+i)(x-i)=x²+1 with synthetic division.

This gives us (x²+2x+1)(x²+1). Now we can use the quadratic formula to find the roots of x²+2x+1.

We get [-2±√(4-4)]/2= -2/2= -1. So -1 is a double root, so this factors as (x+1)²(x²+1).
• One of the zeroes that you got was -1/2 doesn't that mean you have to go 1 left and 2 up on the graph
(1 vote)
• Nopee. That's only when you are using the formula of a slope, because the formula tells you m=y2-y1/x2-x1 (hence, 1 down, 2 to the right).

That the "zero" is -1/2 means that when X= -1/2, your Y-coordinate is going to be "0". Therefore, the point is going to be in the coordinate (-1/2, 0).

Remember that a "zero" or "root" is a point where the graph touches the X-axis.