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# Zeros of polynomials (with factoring): grouping

When a polynomial is given in factored form, we can quickly find its zeros. When its given in expanded form, we can factor it, and then find the zeros! Here is an example of a 3rd degree polynomial we can factor using the method of grouping.

## Want to join the conversation?

• How do we know which do numbers are suppose to be grouped together?
• one thing you could do is order the numbers by degree and then group them together from left to right
(1 vote)
• What would the case be for (x^2-4) ?
• You should eventually learn about the difference of squares, which basically says if you have something like (x^2 - a) you could factor it into (x - sqrt(a)(x - sqrt(a)) so this is really useful if a is a perfect square. It is here in this case.

So (x^2 - 4) = (x-2)(x+2)

Does that help? If you want to know why the rule works just look at what happens if you FOIL (x-a)(x+a)

EDIT

Also works if a number is in front of the x so (ax+b)(ax-b) = (ax)^2 - b^2
• what is the derivative of x
(1 vote)
• The derivative is the slope of the function at a certain position. If the function is at a straight line, the slope will never change. This means that the derivative is a constant. In the save of f(x) = x, the derivative is one. At any x value, the slope of the graph will be one.
• Do you yeet grouping out the math window?
• yeah
(1 vote)
• does anybody know why we need to factor x^2-9 completely, cant u just call an x = 3 and 3^2 - 9 = 0? I supposed u need to find the roots but what really are these roots and why does factoring these to something like (x+b) even become a root?
(1 vote)
• You generally start with a quadratic equation f(x)=x^2-9. This forms a parabola. Without a middle term, we know that the y intercept is the vertex, so setting x=0 we know that the vertex (and y intercept) is (0,-9). If we want to find the x intercepts (zeros, roots, or solutions) we set y=0, so we end up with x^2-9=0. Since this is a difference of perfect squares, we get (x+3)(x-3)=0. Using the 0 product rule, we know either x+3=0 or x=-3 or x-3=0 or x=3. We also note that (-3)^2-9=0 as well as 3^2-9=0.
• Does grouping mean fragmenting an expression(numerical or algebraic) into the products of two factors? OR something else?
(1 vote)
• No, you're referring to factoring. Grouping is a trick that helps with factoring, it is not factoring itself.

Say I have x³+x²-2x-2. No two of these terms have a common factor. However, x³ and x² do have a common factor (of x²) as do -2x and -2 (of -2). Grouping refers to factoring only these sub-expressions, like this
x²(x+1)-2(x+1).

We can then treat this expression as having only two terms with a common factor of x+1, and factor out the x+1 as
(x+1)(x²-2).
• is (x-3)(x+3) the same thing as (x+3)(x-3)?
• Yes, multiplication is commutative, so a*b = b*a. (a and b can be numbers or functions.)
• I only factored to (x+1)(x^2-9) and then did x+1=0, x=-1; and x^2+9=0, x= +/- 3 (i.e. x=-3 or 3). This led to the same answers even though it wasn't completely factored. Is that bad?
(1 vote)
• That's fine. Even if you factor it fully, you would've gotten the same answer.
(1 vote)
• At is it right if I just make the term (x^2 - 9) and equal it to zero as follows:
x^2 - 9 = 0
x^2 = 9
take square roots
x = +3 or -3
(1 vote)
• That is also a correct way to solve those kinds of equations. However, it is also good practice to still being able to know about how to solve difference of squares by factoring.