Question

Will give medal and fan for help
(Algebra 2 Honors Project)
I only have 2 questions left, and all we need to do is answer them, any help would be appreciated :)
-How does the binomial theorem use Pascal’s triangle to expand binomials raised to positive integer powers?
-How is the fundamental theorem of algebra true for quadratic polynomials?

Answer 1

googling fundamental theorem : The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part.

Answer 2

now we know about quadratic polynomials that the number of roots they have depends on the Discriminant ...

when D > 0, we have 2 distinct real roots
when D = 0, we have 1 repeated real root
when D < 0, we have 2 distinct, non-real, complex roots

so the Fundamental theorem holds ... as we have at least 1 root

Answer 3

I'm not sure if I understand how your drawing fits in with the explanation. But I understand what you've written out

Answer 4

the BINOMIAL THEOREM gives us the co-efficients of each term of the expanded form of: \[\Large (a+b)^n\]

Answer 5

for example, for n=4 ... we look at the fifth row of Pascals triangle:\[\Large(a+b)^4=\color{grey}1a^4+\color{green}4a^3b+\color{green}6a^2b^2+\color{green}4ab^3+\color{grey}1b^4\]

Answer 6

So Pascal's triangle helps us to expand binomials by looking at the co-efficients? @PaxPolaris

Answer 7

for larger n ... expanding (a+b)^n can be hard without using the theorem

for (a+b)^3 ... you probably already know it goes 1 3 3 1
for (a+b)^2 ... 1 2 1

Answer 8

would n be the exponent? @PaxPolaris

Answer 9

This video might help explain the second question if you're still having trouble with it:

https://www.khanacademy.org/math/algebra2/polynomial_and_rational/fundamental-theorem-of-algebra/v/fundamental-theorem-algebra-quadratic

Answer 10

@blurbendy the only thing i dont understand about the second question is the "integer powers" part

Answer 11

By second question, I meant "How is the fundamental theorem of algebra true for quadratic polynomials?" I wasn't talking about the binomial one.

Answer 12

\[(a+b)^n\] ... n is a positive integer

Answer 13

Oh, sorry I'm stupid I just glanced at them. No, I understand the fundamental theorem question more or less I think @blurbendy

Answer 14

Okay, good

Answer 15

@blurbendy I watched the video, but it didn't help too much because it mostly covered the stuff I knew. Can you explain in layman's terms for me how the fundamental theorem holds true for quadratic formulas?

Answer 16

the question isnt talking about the quadratic formula, it's talking about quadratic polynomials.

Answer 17

but you can use the quadratic formula to prove that it holds true.

The basic idea is this:

The fundamental theorem states that the degree of a polynomial function is the same as the number of roots.

Quadratics (with x to the second power) have 2 roots. You can use the quadratic formula to find the roots.