<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>
The answer is:
(v-1)(v+1)(v^2+1)
Answer:
25
Step-by-step explanation:
By counting the height ( frequency ) of each block and adding gives the number of students in total
20 - 24 → 5
24 - 28 → 6
28 - 32 → 5
32 - 36 → 2
36 - 40 → 7
Total = 5 + 6 + 5 + 2 + 7 = 25
Answer:
B. load-distance model
Step-by-step explanation:
A. trial and error
Trial and error is "a fundamental method of problem solving. It is characterised by repeated, varied attempts which are continued until success". But this method is not the best in order to compare effectiveness of layouts
B. load-distance model
The load-distance method is a "mathematical model used to evaluate locations based on proximity factors. The objective is to select a location that minimizes the total weighted loads moving into and out of the facility. The distance between two points is expressed by assigning the points to grid coordinates on a map". And that's the correct option since we are trying to measure the effectiveness of layouts quantitatively.
C. exponential smoothing
This is "a rule of thumb technique for smoothing time series data using the exponential window function". Wheighting observations using the exponential function. But this is a techinique used to smooth s time series not to compare effectiveness of layouts.
D.process control charts
The Control Chart is a "graph used to study how a process changes over time with data plotted in time order". But we don't want to see how the process changes the objective is quantitatively compare the effectiveness of layouts, and this one is not the best option for this.
E. mean absolute deviation (MAD)
The median absolute deviation(MAD) is "a robust measure of how spread out a set of data is. The variance and standard deviation are also measures of spread, but they are more affected by extremely high or extremely low values and non normality". But again is just a measure of spread and not allow to compare effectiveness of layouts.
9514 1404 393
Answer:
17. 5
18. 17
Step-by-step explanation:
The distance formula is used for the purpose.
d = √((x2 -x1)² +(y2 -y1)²)
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17. d = √((3-6)² +(1-5)²) = √((-3)² +(-4)²) = √(9+16) = √25 = 5
The distance between the points is 5 units.
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18. d = √((-1-7)² +(12-(-3))²) = √(64 +225) = √289 = 17
The distance between the points is 17 units.
