Will mark brainliest consider the polynomial expression below x^3-3x^2+81x-243 rewrite the polynomial in the form (x-d)(x-e)(x+f) where d is a real number, and e and f are complex numbers in the form bi

(x - 3)(x - 9i)(x + 9i)

Step-by-step explanation:

note that x = 3 gives

3³ - 3(3)² + 81(3) - 243 = 27 - 27 + 243 - 243 = 0, hence

x = 3 is a root and (x - 3) is a factor

x³ - 3x² + 81x - 243 ÷ (x - 3)

= (x - 3)(x² + 81)

solve x² + 81 = 0 ⇒ x² = - 81 ⇒ x = ± 9i, thus

x³ - 3x² + 81x - 243 = (x - 3)(x - 9i)(x + 9i) ← in factored form

x³ - 3x² + 81x - 243  = ( x - 3 )( x - 9i )( x + 9i )

Step-by-step explanation:

Given Expression is x³ - 3x² + 81x - 243

We need to factorize the given expression in form of ( x - d )( x - e )( x + f ) where d is a real number, and e and f are complex numbers in the form bi.

From Rational factor theorem,

Factors of 243 is ±1 , ±3 , ±9 , ±27 , ±81 , ±243.

Now by hit and trial we find first factor of real number.

let, p(x) = x³ - 3x² + 81x - 243

put x = 3

p(3) = 3³ - 3 × 3² + 81 × 3 - 243  = 27 - 27 + 243 - 243 = 0

So, The first factor is ( x - 3 ).

Now to find other factor we divide given polynomial with x - 3.

Division is attached.

We get Quotient = x² + 81

Remainder = 0

Now, Factorize x² + 81 = ( x )² - ( 9i )² = ( x - 9i )( x + 9i )

Therefore, x³ - 3x² + 81x - 243  = ( x - 3 )( x - 9i )( x + 9i )

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

3x + 4y -25 =0 represents the amount of change sianna should recieve