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Tamiku [17]
3 years ago
14

What is the coefficient of xy^2 in the expansion of (x+y)^3?

Mathematics
2 answers:
bija089 [108]3 years ago
6 0

Answer:

The coefficient of the xy² term is 3.

For your question about xy⁵, the coefficient is 6

And for the x²y² question, it is also 6

Step-by-step explanation:

We can simply expand the expression to find the answer:

(x + y)³

= (x + y)(x + y)(x + y)

= (x + y)(x² + 2xy + y²)

= x³ + 2x²y + xy² + x²y + 2xy² + y³

= x³ + 3x²y + 3xy² + y³

So the term term containing xy² has a coefficient of 3

Now, there's actually a simple way to do this, and that's using Pascal's triangle.  If you're not familiar with it, here's a quick representation of the first few lines:

0                                    1

1                                  1     1

2                              1     2     1

3                           1     3     3     1

4                       1      4     6     4     1

5                    1      5    10    10    5    1

6                1      6     15   20   15    6    1

7             1      7     21   35   35   21   7    1

8         1      8     28   56  70   56   28  8   1

You'll notice that each number is simply the sum of the two numbers above it; below 4 and 6 wes see 10, below 20 and 15 is 35, etc.

This is very useful in this case, as the you may notice that the solution to each of the questions asked can be answered by this triangle.  For instance, when we solved (x + y)³, the coefficients of the terms were  1, 3, 3, and 1.  For a perfect square, it's 1, 2 and 1, and so on.  So to get the answers to your questions, you need only look at this triangle.

For example, want the the coefficent of the fourth term in the expansion of (x + y)⁷, it's easy, just look at the seventh row, and look for the fourth number, which would be 35.

But as we're given the x and y powers, but not the position of the term, we need to work that out as well.  Thankfully there's an easy trick to that.  The exponents will always be like this

x^{n}y^0 + x^{n - 1}y^1 + x^{n - 2}y^2 + ..... + x^2y^{n - 2} + x^1y^{n - 1} + x^0y^{n}

where n is the number we're raising the binomial to.

We could put together an equation to solve this, but instead of going through a whole lesson, I'll just update the answer at the top.

Basile [38]3 years ago
5 0

Step-by-step explanation:

Using Binomial Expansion,

(x + y)³

= 3C0 * x³ + 3C1 * x²y + 3C2 * xy² + 3C3 * y³.

Therefore the coefficient of xy² is 3C2 = 3.

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for the graph y= 12 find the slope of a line that is perpendicular to it and the slope of a line parallel to it. Explain your an
ra1l [238]

y = 12 is a line in which every point has a y-coordinate of 12 no matter what the x-coordinate is. That means that y = 12 is a horizontal line. The equation of line y = 12 can be written as y = 0x + 12 clearly showing m = 0 (slope is zero) and the y-intercept is 12. A line parallel to line y = 12 has the same slope, so its slope is also zero.

A line perpendicular to y = 12 is a vertical line. For a vertical line, the slope is undefined because it involves division by zero which is undefined in math.

4 0
2 years ago
0.2(3k+1)=-0.4-(-0.2-0.1K)
Tanya [424]

Answer:

k=−0.8

Step-by-step explanation:

Start with the equation: 0.2(3k+1)=-0.4-(-0.2-0.1K)

Use distributive property on each side of the equation. For the left side, distribute 0.2 into 3k and 1, and for the right side, distribute -1 into -0.2 and -0.1k. (We use a -1 because the negative sign on the outside of the parenthesis is equivalent to -1)

(0.2)(3k) + (0.2)(1) = 0.6k + 0.2

(-1)(-0.2) + (-1)(-0.1k) = 0.2 + 0.1k

New equation: 0.6k + 0.2 = -0.4 + 0.2 + 0.1k

Now we can combine like terms on the right side. (-0.4 and 0.2)

0.6k + 0.2 = 0.1k - 0.2

Now to get rid of k on one side, we can subtract 0.1k from the right side.

(0.6k - 0.1k) + 0.2 = (0.1k - 0.1k) - 0.2

0.5k + 0.2 = -0.2

Now we need to get 0.5k by itself, so we'll subtract the 0.2 on both sides.

0.5k + (0.2 - 0.2) = (-0.2 - 0.2)

0.5k = -0.4

To get k by itself, we must divide both sides by 0.5.

(0.5k / 0.5) = (-0.4 / 0.5)

k = -0.8

4 0
2 years ago
Find the equation of a line that is PERPENDICULAR to y = 1/3 x -2 & passes through the point ( 3, 2 ). (CHECK PIC FOR ANSWER
densk [106]

Answer:

The answer to your question is the letter b)    y = -3x + 11

Step-by-step explanation:

Data

Original line  y = 1/3x - 2

Point = (3, 2)

New line = ? (perpendicular)

Process

1.- Get the slope of the original line

The slope is the coefficient of the x in the line equation.

       y = 1/3x - 2

       slope = m = 1/3

2.- Find the slope of the new line

       new slope = -3  (because the lines are perpendicular)

3.- Find the equation of the new line

         y - y1 = m(x - x1)

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-Substitution

        y - 2 = -3(x - 3)

-Simplification

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        y = -3x + 9 + 2

-Result

       y = -3x + 11

         

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Nezavi [6.7K]
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Answer: x = 14y + 65
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