Question

what should be subtracted from 9x⁶-12x⁵+4x⁴-18x³-12x²+18 to make it a perfect square​

Answer 1

Answer:

i

don't

know

because

it

is

very

hard

Answer 2

Answer:

24x² + 9 should be subtracted from 9x⁶-12x⁵+4x⁴-18x³-12x²+18 to make it a perfect square.

Step-by-step explanation:

First, observe that tge first term,9x⁶ is already3x³squared.So we could construct a cubic polynomial that looks like this:

$P(x) = 3 {x}^{3} + a {x}^{2} + bx + c$

Then find what needs to be subtracted from the expression to make it

$= {P(x)}^{2}$

$(3 {x}^{3} + a {x}^{2} + bx + c)(3 {x}^{3} + a {x}^{2} + bx + c) - 12 {x}^{5} + 4 {x}^{4} - 18 {x}^{3} - 12 {x}^{2} + 9 = 9$

So from that we can work out the following equations for the x⁵,x⁴,x³ terms,

$6a = - 12 \\ 6b + {a}^{2} = 4 \\ 6c + 2ab = - 18$

From there we can easily see

$a = - 2 \\ b = 0 \\ c = - 3$

Then we have three more equations for the x²,x and constant terms,

$2ac + {b}^{2} = - 12 - k₂ \\ 2bc=0-k₁ \\ {c}^{2} = 18 - k₀$

Where K₂,k₁,k₀ are coefficients of the polynomial that needs to be subtracted from the polynomial subtracted from the expression to make the whole square o given expression .

Therefore −24x² + 9 from the expression will produce an expression.