Answer:
When we sum polynomials, like:
p(x) = a*x^2 + b*x + c
q(x) = k*x^3 + n*x^2 + m*x + ñ
The sum grups terms with the same power of x, where we just take the x part as a common factor, so we can write, in this case:
p(x) + q(x) = k*x^3 + (a + n)*x^2 + (b + m)*x + (c + ñ)
This is just an example, let's see how we can apply this to the given problems.
a) We start with:
3*x^2 - 2*x + 1
and we want to add something to get: 3 + 5*x - 7*x^2
Then we need to add a quadratic polynomial a*x^2 + b*x + c
(we know this because the end polynomial is also a quadratic one, the same as the first one)
Then we get
(3*x^2 - 2*x + 1) + (a*x^2 + b*x + c) = -7*x^2 + 5*x + 3
(3 + a)*x^2 + (-2 + b)*x + (1 + c) = -7*x^2 + 5*x + 3
Is easy to see that we must have:
3 + a = -7
-2 + b = 5
1 + c = 3
Solving these 3 equations, we get:
a = -7 - 3 = -10
b = 5 + 2 = 7
c = 3 - 1 = 2
Then the polynomial that we must add is:
-10*x^2 + 7*x + 2
b)
Here we start with:
(-4*m^2 + 5*m - 3)
and want to subtract something to get: m^2 - 3*m
Then let's subtract a polynomial like:
a*m^2 + b*m + c
And let's do the same than in the case "a"
(-4*m^2 + 5*m - 3) - (a*m^2 + b*m + c) = m^2 - 3*m + 0
(-4 - a)*m^2 + (5 - b)*m + (-3 - c) = m^2 - 3*m + 0
Then we must have:
-4 -a = 1
5 - b = -3
-3 - c = 0
Solving the above equations, we get:
a = -4 - 1 = -5
b = 5 + 3 = 8
c = -3
Then the polynomial that we must subtract is:
-5*m^2 + 8*m - 3
c)
Here we have a lot of variables, so remember to take the correct common factors, take your time:
we start with:
(4*b^2 + 5*b*c) + (-2*b^2 - 2*b*c - 2*z^2) + (2*b*c - 4*c)
We want to subtract the sum:
(5*b^2 - c^2) + (-3*b^2 + 2*b*c + c)
First let's simplify both of these sums, we can rewrite the first one as:
(4 - 2)*b^2 + (5 - 2 + 2)*b*c - 2*z^2 - 4*c
= 2*b^2 + 5*b*c - 2*z^2 - 4*c
Now, for the other sum we can simplify as:
(5 - 3)*b^2 - c^2 + 2*b*c + c
= 2*b^2 - c^2 + 2*b*c + c
Finally, we can calculate the difference between these two as:
( 2*b^2 + 5*b*c - 2*z^2 - 4*c) - (2*b^2 - c^2 + 2*b*c + c)
This is equal to:
(2 - 2)*b^2 + (5 - 2)*b*c - 2*z^2 + (-4 - 1)*c - c^2
3*b*c - 2*z^2 - 5*c + c^2