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What is 28a11b7 divided by 4a3b?
12
What do you mean by polynomials?
A polynomial is any expression (i.e. no = sign) that is the sum of several monomials. Subtraction is ok, but to be a polynomial they can't be divided, and they can't be multiplied with parentheses. Polynomials: 5x+4xy; x2+3x-2; 42x-1. Not Polynomials: (10x)/2+4xy; x(x+3); 45. ---- A monomial is one or more numbers or variables multiplied together. For example, 5x, 23, x2, and 4a3b are monomials. The exponents must be natural numbers.
How is Pascal's triangle related to the Binomial Theorem?
I think an example will help most people see it better than just an explanation/answer. So first a few examples are presented and than a general answer.Start with (1+x)2 = 1+2x+x2 and look at the coefficients of the results you will see that they are 1, 2, 1. Now do it for (1+x)3 and they are 1, 3, 3, 1. These, of course, are the lines from Pascal's Triangle. I put the first part of the triangle below (in left-justified form). You should notice that when the exponent is 2, we use the third line and when the exponent is 3, we use the fourth line of the triangle.11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 11 6 15 20 15 6 11 7 21 35 35 21 7 11 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1In case you wonder about the first two rows. Look at (1+x)0 and(1+x)1, their coefficients come from those two rows.Now look at (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 which is more general and of course the coefficients come from the 5th line of the triangle.So the answer to the question is that if we look at the binomial (a + b)nThe n+1 row of Pascal's triangle gives us the coefficients of the expanded form of the binomial. Seeing the examples first often makes this easier to see and understand. It is the n+1 row because the first row of the triangle is any binomial with an exponent of 0.
