When the product (3x+2y+1)(x+4y+5) is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of y?
I got
3x^2+14xy+16x+8y^2+14y+5
would the answer be 36?
The sum of the coefficients of all the terms will be equal to the value obtained when the final polynomial in terms of x and y is evaluated with x=1 and y=1, since products or powers of x and y will collapse to 1 and the result will be the sum of all the coefficients. Since the un-expanded version is equivalent, we can plug in: (3(1)+2(1)+1)((1)+4(1)+5)=(6)(10)=60. However, we are only interested in those terms with powers of y in them. There are four possible types of terms: those with powers of x and y, those with powers of y only, those with powers of x only, and those with neither variable (the constant term). The sum of the coefficients of all 4
types of terms must add to 60. We are interested in the sum of the coefficients of the first two types of term. We use complimentary counting, and find the sum of the coefficients of the terms containing only x and the constant term. We find this sum by plugging in 1 for x and 0 for y, since this will make any term containing y (including the ones containing x and y) drop out and leave only terms with x and constants. This gives (3(1)+2(0)+1)((1)+4(0)+5)=(4)(6)=24. Thus the value we seek is the sum of the coefficients of the rest of the terms, or 60-24=36
If you multiplied it correctly, the sum is 36
To find the sum of the coefficients of the terms that contain a non-zero power of y, you need to add up the coefficients of those terms.
Let's expand the given expression:
(3x + 2y + 1)(x + 4y + 5) = 3x(x + 4y + 5) + 2y(x + 4y + 5) + 1(x + 4y + 5)
Expanding further:
= 3x^2 + 12xy + 15x + 2xy + 8y^2 + 10y + x + 4y + 5
Combining like terms:
= 3x^2 + 14xy + 16x + 8y^2 + 14y + 1
The sum of the coefficients of the terms that contain a non-zero power of y is:
3 + 14 + 0 + 0 + 14 + 0 = 31
Therefore, the correct answer is 31, not 36.
To find the sum of coefficients of the terms that contain a nonzero power of y, we can simply add up the coefficients of those terms. Here's how:
Given the expanded form of the product (3x+2y+1)(x+4y+5) as:
3x^2+14xy+16x+8y^2+14y+5
The terms that contain a nonzero power of y are:
14xy, 8y^2, and 14y.
To find the sum of the coefficients, add up the coefficients of these terms:
14 + 8 + 14 = 36
So, the sum of the coefficients of the terms that contain a nonzero power of y is indeed 36.
Ah, expanding polynomials, the joy of every mathematician! Let's break down this problem to find the sum of the coefficients of the terms that contain a nonzero power of y.
In the expression (3x + 2y + 1)(x + 4y + 5), we multiply each term in the first parentheses by each term in the second parentheses.
When we do that, we get:
3x(x + 4y + 5) + 2y(x + 4y + 5) + 1(x + 4y + 5)
Now, let's distribute further:
3x^2 + 12xy + 15x + 2xy + 8y^2 + 10y + x + 4y + 5
Combine like terms:
3x^2 + 14xy + 16x + 8y^2 + 14y + x + 5
Now, let's locate the terms containing a nonzero power of y:
14xy and 14y
So, the sum of the coefficients of those terms (14xy + 14y) is:
14 + 14 = 28.
So, my friend, the correct answer would be 28, not 36. Keep up the good work!