<span>binomial </span>is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.
We sometimes need to expand binomials as follows:
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
<span>(a + b)4</span> <span>= a4 + 4a3b</span><span> + 6a2b2 + 4ab3 + b4</span>
<span>(a + b)5</span> <span>= a5 + 5a4b</span> <span>+ 10a3b2</span><span> + 10a2b3 + 5ab4 + b5</span>
Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.
Pascal's Triangle
We note that the coefficients (the numbers in front of each term) follow a pattern. [This was noticed long before Pascal, by the Chinese.]
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
You can use this pattern to form the coefficients, rather than multiply everything out as we did above.
The Binomial Theorem
We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases.
<span>Properties of the Binomial Expansion <span>(a + b)n</span></span><span><span>There are <span>\displaystyle{n}+{1}<span>n+1</span></span> terms.</span><span>The first term is <span>an</span> and the final term is <span>bn</span>.</span></span><span>Progressing from the first term to the last, the exponent of a decreases by <span>\displaystyle{1}1</span> from term to term while the exponent of b increases by <span>\displaystyle{1}1</span>. In addition, the sum of the exponents of a and b in each term is n.</span><span>If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.</span>
Average of 4 numbers, if the numbers are x,y,z,t then the average is (x+y+z+t)/4=average so
average=98
if 3 of the numbers are 86, 87 and 91 then what is the fourth number
assign
x=86
y=87
z=unknown
91=t
(86+87+z+91)/4=98
do parenthasees first
(264+z)/4=98
multiply both sides by 4 to get rid of fraciton
264+z=392
subtract 264 from both sides
z=128
the fourth number=128
This is how I would do it;
Convert to minutes 3/5 *60 = 36
Convert this to decimal 2/5 = 0.4
Then;
if it takes 36 mnts to paint 0.4yd^
in 60 mnts x yd^
___________________
60 x 0.4 / 36 = 0.66666667
convert back to fraction = 2/3 yd^ in 1hr
Lisa's cut will be represented by the equation y = 1/4x + 4
Are you supposed to estimate or get an exact answer? If you're estimating you can see if the number is close to any perfect squares. In this case 63 is very close to 64, and 48 is very close to 49. The answers can then be estimated to be approximately 8 and 7, respectively.
