Expand the following using the binomial theorem and pascal’s triangle. (x + 2)6 (x − 4)4 (2x + 3)5 (2x − 3y)4 in the expansion of (3a + 4b)8, which of the following are possible variable terms? explain your reasoning. a2b3; a5b3; ab8; b8; a4b4; a8; ab7; a6b5

The possible terms are .

Step-by-step explanation:

Binomial Theorem:

Using Binomial Theorem, we get

Similarly,

The Pascal’s triangle is given below. In Pascal’s triangle nth row represents the coefficients of the expression of .

In the expression , a=x, b=2 and n=6. Using 7th row of Pascal’s triangle we get

Similarly,

In a=x, b=-4 and n=4. Using 5th row of Pascal’s triangle we get

In a=2x, b=3 and n=5. Using 6th row of Pascal’s triangle we get

In a=2x, b=-3y and n=4. Using 5th row of Pascal’s triangle we get

In binomial expansion of the degree of each term must be 8. It means the sum of powers of a and b is equal to 8.

For the dgree of term is 2+3=5. So, it is not possible variable term.

Therefore the possible terms are .

let soccer = x and golf = y

part a

1st equation: x + y = 125

2nd equation: x = y +45

part b

replace x in the 1st equation with the 2nd one:

y + 45 + y = 125

simplify:

2y + 45 = 125

subtract 45 from each side:

2y = 80

divide both sides by 2:

y = 80/2

y = 40

she plays golf for 40 minutes.

part c

no because she plays soccer for 45 minutes longer than golf 40 +45 =   85 minutes, not 80 minutes.