help with math problem.? PRECALCULUS?

Question

find the 5th term of (1-y)^8

what is it asking for?
is there a formula?
step by step explanations would be very helpful!

Answer 1

There are several ways to approach this.

One would be to simply start expanding:
(1-y)(1-y)(1-y)^6 = (1-2y+y^2)(1-y)^6=(1-3y+3y^2-y^3)(1-y)^5...
expanding terms one by one.

Another would be to square (1-y), square the result, and square it again:
(1 - y)^2 = 1 - 2y + y^2
(1 - 2y + y^2)^2 = 1 - 4 y + 6 y^2 - 4 y^3 + y^4
that whole thing squared is:
1 - 8 y + 28 y^2 - 56 y^3 + 70 y^4 - 56 y^5 + 28 y^6 - 8 y^7 + y^8
We are lucky that it's an 8, since squared-squared-squared is the same as the eight power.

Another way would be to "look it up" in Pascal's triangle - but that won't help you understand anything unless you understand what's going on there - which is the fourth (and final) method I'll explain:

Binomial coefficients? What are those? Well, it's the number of ways to choose k things out of n things. Here, we're choosing k terms out of 8 factors. You see, imagine we're trying to pick a term out of the following:

(1-y)(1-y)(1-y)(1-y)(1-y)(1-y)(1-y)(1-y)

We want the constant term first. How many ways can things in these multiply to be constants? Only if we multiply the 1s together (any of the -y terms will give us a y). So the constant term is (8 CHOOSE 0) * 1^8 = 1.

Now we want the y term. We can pick a -y term from any of these - there are 8 to choose from. So we get (8 CHOOSE 1) 1^7 * (-y) = -8y

You see, when you expand this expression out, you can think of it as adding up every possible choice of terms from the factors - in the first, you can choose 1 or -y, in the second you can choose 1 or -y, in the third etc... The only way to get a constant is to choose all 1s. There are 8 ways to get a single y, depending on which factor you get the y from. There are 8*7 / 2 = (8 CHOOSE 2) ways to pick out two y terms for the y^2 term - since you pick out of 8 for the first y, and out of the remaining 7. We divide by 2 because the order of our choices does not matter - we can choose the first and second y, or the second and first, it's still the same.

So these binomial coefficients can be computed in a number of ways, one of which is Pascal's triangle (see source below), which gives us a quick idea of what the coefficients look like for (1+y)^8 -- look at the ninth row. HOWEVER because we have -y, we have to add an alternating +/- pattern in the answer, because when we pick an odd number of ys, we get an odd number of - signs multiplying together. See the second and third sources, which are about binomial expansions.

Answer 2

The question is asking you to find the fifth term if you were to expand the expression.
You can do this by the binomial theorem.
Pascal's triangle can help find the coefficients.
The numbers in pascal's triangle come from combination formula.

Steps for expanding a binomial in general.

Step one: The first term decreases in power. Write the second term increases in power.
(If the sign in the factor is postive all the signs are positive)
However if the sign is negative in the binomal expression
then the sign is negative)
In you question the binomal (1-y) has a minus sign so the
sign alternates in the expression.

1^8y^0 - 1^7y^1 + 1^6y^2 - 1^5y^3 + 1^4y^4 - 1^3y^5 + 1^2y^6 - 1^1y^7 + y^81^0

Step 2 : The coefficients of the expansion are from the combination formula where a combination is (nCr)
with n equal to the power that the binomial is expanded and r increase with the term (r starts at zero)

8C0 * (1^8y^0) - 8C1 (1^7y^1 ) + 8C2* (1^6y^2) -
8C3* (1^5y^3) + 8C4*(1^4y^4) - 8C5 *(1^3y^5 )
+8C6*(1^2y^6) - 8C7*(1^1y^7) + 8C8 *(y^81^0)

Answer 3

If a rectangle has a perimeter of 14 cm, its four sides add to 14. Alternatively, since it has two different side lengths, 2 times the sum of those lengths is also 14. Let x be one of the sides, and L be the other. We know that x+x+L+L = 2x + 2L = 14 Dividing both sides by 2 gives us x + L = 7 Isolate L by moving x to the right side: L = 7 - x So, now we have both side lengths of the rectangle in terms of x: x, and 7-x. The area of a rectangle is the product of its side lengths, which, in this case, would be x*L = x*(7-x) = x*7 - x*x = 7x - x^2. So our function is A(x) = 7x - x^2. This is an upside-down parabola, so the maximum area would be the value of A(x) at the vertex of the parabola. If the equation is written as ax^2 + bx + c, the x-coordinate of the vertex is at -b/2a. So, substituting our values of b and a gives us x = -7/(2*-1) = -7/-2 = 7/2 Plugging this into A(x) to get the area gives us A(7/2) = 7*7/2 - (7/2)^2 = 49/2 - 49/4 = 49/4 cm^2 as the maximum area.

Answer 4

If you expand the expression (1-y)^8, which means to multiply it by itself 8 times, you get a polynomial with 9 terms with powers of y up to y^8. If you arrange the terms in descending powers of y, it looks like this:
y^8 -8y^7 + 28y^6-56y^5 +70y^4 .... etc.
The fifth term is 70y^4.

Answer 5

Binomial theorem:
(8C4)(1^4)(-y^4)=70y^4