(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

last one.

step-by-step explanation:

last one.

generally speaking, the others follow some sort of pattern that correlates to either linear (looks like a straight line) or sinusoidal (looks like an s).

the last one is just a random cloud of points that doesn't correlate to any particular known order.

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