Decompose in factors of first grade the polynomial -2x^4-8x^3+10x^2+72x+72.

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to convert the standard form of the given polynomial into its factored form, hence, you need to look for the roots of polynomial. You may look for the roots of polynomial among the values obtained dividing the divisors of  constant term 72, to divisors of leading coefficient -2, such that:

Possible roots = `(+-1)/(+-1), (+-1)/(+-2), (+-2)/(+-1), (+-2)/(+-2), (+-3)/(+-1),(+-3)/(+-2), (+-4)/(+-1),(+-4)/(+-2), (+-6)/(+-1),(+-6)/(+-2), (+-8)/(+-1),(+-8)/(+-2),(+-12)/(+-1),(+-12)/(+-2),(+-18)/(+-1),(+-18)/(+-2),(+-24)/(+-1), (+-24)/(+-2),(+-36)/(+-1),(+-36)/(+-2),(+-72)/(+-1),(+-72)/(+-2)`

You need to test the values above in equation, such that:

`-2-8+10+72+72 != 0 (x = 1)`

`-2+8+10-72+72 != 0 (x = -1)`

`-2*2^4-8*2^3+10*2^2+72*2+72 = -32 - 64 + 40 + 144 + 72 != 0 (x = 2)`

`-2*(-2)^4-8(-2)^3+10(-2)^2+72(-2)+72 = -32 + 64 + 40 - 144 + 72 = 0 => x = -2` is one root

You need to continue test if x = 3 is a root for polynomial, such that:

`-2*3^4-8*3^3+10*3^2+72*3+72 = -162 - 216 + 90 + 216 + 72 = 0 => x = 3` is the second root.

You may write the factored form of polynomial, such that:

`-2x^4-8x^3+10x^2+72x+72 = -2(x - x_1)(x - x_2)(x - x_3)(x - x_4)`

Since `x_1 = -2` and `x_2 = 3` yields:

`-2x^4-8x^3+10x^2+72x+72 = -2(x + 2)(x - 3)(x - x_3)(x - x_4)`

`-2x^4-8x^3+10x^2+72x+72 = -2(x + 2)(x - 3)(ax^2 + bx + c)`

You need to continue test if `x = -3` is a root for polynomial, such that:

`-2*(-3)^4-8*(-3)^3+10*(-3)^2+72*(-3)+72 = -162 + 216 + 90 - 216 + 72 = 0 => x = -3` is the third root

`-2x^4-8x^3+10x^2+72x+72 = -2(x + 2)(x - 3)(x + 3)(x - x_4)`

`-2x^4-8x^3+10x^2+72x+72 = -2(x + 2)(x^2 - 9)(x - x_4)`

`-2x^4-8x^3+10x^2+72x+72 = (-2x^3 + 18x - 4x^2 + 36)(x - x_4)`

`-2x^4-8x^3+10x^2+72x+72 = -2x^4 + 2x^3*x_4 + 18x^2 - 18x*x_4 - 4x^3 + 4x^2*x_4 + 36x - 36x_4`

Equating coefficients of like powers yields:

`72 = - 36x_4 => x_4 = -72/36 => x_4 = -2`

Hence, the root `x = -2` is of order of multiplicity 2.

`-2x^4-8x^3+10x^2+72x+72 = -2(x + 2)(x - 3)(x + 3)(x - 2)`

`-2x^4-8x^3+10x^2+72x+72 = -2(x^2 - 9)(x + 2)^2`

Hence, evaluating the factored form of the given polynomial yields `-2x^4-8x^3+10x^2+72x+72 = -2(x^2 - 9)(x + 2)^2` .

 

justaguide eNotes educator| Certified Educator

The polynomial -2x^4-8x^3+10x^2+72x+72 has to be factorized.

-2x^4-8x^3+10x^2+72x+72

= -2*(x^4 + 4x^3 - 5x^2 - 36x - 36)

= -2*(x^4  - 9x^2 + 4x^2  - 36 + 4x^3 - 36x)

= -2*(x^2(x^2 - 9) + 4(x^2 - 9) + 4x(x^2 - 9))

= -2*(x^2 + 4x + 4)(x^2 - 9))

= -2*(x + 2)^2(x - 3)(x + 3)

The polynomial -2x^4-8x^3+10x^2+72x+72 = -2*(x + 2)^2(x - 3)(x + 3)