(a) What must be added to 1 - 2x + 3x² to obtain 3 + 5x - 7x2? (b) What must be subtracted from -4m2 + 5m - 3 to obtain m2 – 3m?
(c) From the sum of 4b2 + 5bc, -2b2 - 2bc – 2z2 and 2bc + 4c?, subtract the sum of 5b²-c²and -3b²+2bc+c
solve step by step ​

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

x=17

step-by-step explanation:

solve 3x-24=x+10

then that makes it 2x=34 because you subtract an x from both sides to get rid of the one on the right, and then you add 24 to both sides to get rid of the 24 on the left

divide by both sides by 2 to get your answer so that you can get your answer

what is

step-by-step explanation: