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## Integrated math 3

### Course: Integrated math 3>Unit 2

Lesson 4: Factoring higher degree polynomials

# Factoring higher degree polynomials

Factoring higher degree polynomials involves breaking down complex expressions into simpler parts. This process includes identifying common factors, using the distributive property, and recognizing perfect squares. Sal demonstrates how to factor a partially factored polynomial and how to factor a third degree polynomial by grouping.

## Want to join the conversation?

• What are some common real world applications for this? •   You may need to use factoring often if you have a real world job. If you decide to become an economist, statistician, engineer, mathematician, or any kind of physical scientist. You would commonly use factoring. For example, data containing complex algebraic fractions can look extremely difficult. You would need to factor in order to simplify and make use of the data in an easier way. Basically, it will make your life more simple though it seems annoying right now. Hope this helps!
• So I have watched video after video and I also took notes. •  The first example is set up like this:
(6x^2 + 9x)(x^2 - 4x +4).
What Sal did was take the GCF out of each set of parentheses. As you may have seen in previous videos in this unit, the way to find the GCF from the left set of parentheses is to find the GCF of the coefficients. In this case, the GCF of 6 and 9 is 3. The next step to find the GCF of the full terms is to look at the variables. There is an x^2 and an x on its own. Let's list out the factors:
x^2 factors are 1 and x^2, and x and x since those numbers multiply to get to x^2.
x factors are x and 1.
We can see by looking at the factors that the greatest factor is x (assuming x is greater than one, but we aren't going to worry about that because we don't have to solve for x in this problem). Since the GCF of the variables is x and the GCF of the coefficients is 3, we multiply them together to get 3x.
Now that we have our GCF for the left set of parentheses, we can divide everything in the left set by our GCF, and bring the GCF out of the parentheses. This is better illustrated in the video Taking common factor from binomial, under Taking common factors earlier in this unit, so I am going to skip explaining that part.
(6x^2 + 9x)/3x is (2x + 3). Now we bring the 3x to the outside of the parentheses to get 3x(2x + 3).
That's one half of the equation. The other we can tell just by looking that it is a perfect square, so we split it apart as shown in the first unit called Polynomial Arithmetic, with the video Polynomial special products: perfect square.
Splitting (x^2 - 4x + 4) into its square roots results in this:
(x - 2)(x - 2).
The next step is to put all of that together. This gets us
3x(2x + 3)(x - 2)(x - 2) Since you can no longer factor this equation, it is in simplest form. That means we just leave it like that.

The second example is a little different:
x^3 - 4x^2 + 6x - 24.
The easiest way to solve this is to factor by grouping. To do that, you put parentheses around the first two terms and the second two terms.
(x^3 - 4x^2) + (6x - 24). Now we take out the GCF from both equations and move it to the outside of the parentheses.
x^2(x - 4) + 6(x - 4). As you can see, the sets of numbers inside the parentheses are the same. This means that we can take the numbers outside the parentheses and put them in their own set.
(x^2 + 6)(x + 4). When you multiply that out, you get x^3 - 4x^2 + 6x + 24. That means that this is as simplified as you can get your equation. Also, it means you just did all of that math to get a circle (start in one place, end in the same).
• I'm still confused on this still after watching an hour. • At he mentions there are other videos that he assumes we've already seen. Which videos are they? • is there another way other than "Grouping" to solve the higher-degree polynomials?
(1 vote) • Yes, there are several methods to solve higher-degree polynomials (polynomials of degree three or higher) other than grouping. The most common methods include:

1. *Factoring*: This method involves factoring the polynomial into simpler expressions that can be set to zero to find the roots (solutions). Factoring works well for polynomials with rational roots or when they can be factored into binomial or trinomial expressions. However, it becomes more challenging for higher-degree polynomials with no simple factorization.

2. *Root-Finding Algorithms*: For polynomials that cannot be easily factored, numerical root-finding algorithms like the Newton-Raphson method, bisection method, or the secant method can be used to approximate the roots. These methods iteratively refine an initial guess until an accurate root is found.

3. *Completing the Square*: This method is commonly used for solving quadratic equations (polynomials of degree 2) but can be extended to higher-degree polynomials in some cases. By completing the square, you can transform certain higher-degree polynomials into a quadratic form that can then be solved using the quadratic formula.

4. *Descartes' Rule of Signs*: This rule provides information about the possible number of positive and negative real roots of a polynomial by counting the sign changes in its coefficients. It doesn't directly give the roots but narrows down the possibilities.

5. *Graphical Methods*: For a rough estimation of the roots, you can plot the graph of the polynomial and identify the approximate values where it crosses the x-axis, representing the roots.

6. *Numerical Methods*: Besides root-finding algorithms, there are other numerical methods like polynomial interpolation and numerical integration techniques that can help approximate the roots of higher-degree polynomials.

It's essential to note that there is no general algebraic formula (analogous to the quadratic formula for degree 2 polynomials) for finding roots of polynomials with degrees higher than 4. For these higher degrees, numerical methods are often used to approximate the roots. Additionally, for some higher-degree polynomials, finding exact solutions may not be feasible or practical, and numerical methods are the best approach.
• I stopped the video and took a crack at it and got your answer, but then I also decided just for, um, fun? to recombine them to get 6x^4-15x^3-12x^2+36x, then I saw i could factor back out 3x, so 3x(2x^3-5x^2-6x+12)... which I noticed 3x was both in the original answer 3x(2x+3)(x-2)^2... so I presume that the 2x^3-5x^2-6x+12 could (or would) factor back down into (2x+3)(x-2)^2.... but i'm sort of lost on this circle... is only it only considered LCM in one of those, both of those? (neither?)... i guess i just noticed we weren't technically looking for LCM, so maybe this was just an exercise in options?

I feel like guy who's mastered the art of stapling only to staple all four corners of a stack of papers. =P • Is anyone else confused about this concept?! As a hormonal teenager this is making me feel a little on edge and kind of frustrated. Explanation please!!😶 • I just literally don't understand how he got the (x-2) (x-2) part? I just am watching all the videos I can and still don't understand polynomials... pls, help...   