how to solve 5 degree polynomial

Question

x^5-2x^4+x^3+x+5

Answer 1

x⁵ - 2x⁴ + x³ + x + 5

Note:  In order to do synthetic division to factor a polynomial

we need to have all the coefficients in order from highest.. 5 ..

down to zero ... note that 5 = 5x⁰

 

Best is to rewrite it as: x⁵ - 2x⁴ + x³ + 0x² + x + 5

You see the coefficients are 1, -2, 1, 0, 1, 5

 

The roots of the polynomial will come from the factors

to the constant.... 5... divided by the factors of the the

coefficient of the highest variable with highest exponent .. 1

 

The roots will be found in ±{1, 5}/±{1}  so 1, -1, 5, -5

 

If -1 is a root then (x+1) is a factor

 

Set up the synthetic division as below

 

-1  |  1    -2    1    0    1   5        bring the 1st coefficient down.

     |        -1    3   -4    4  -5       Multiply the possible root by it and

     ---------------------------       carry up under next coefficient.

       1      -3   4   -4    5   0       Add down and repeat process.

 

If the last sum is zero, as it is in this case, then the root IS an

actual root.   The numbers in the bottom row, the sums, form

the coefficients of the other factor.

This means that

x⁵ - 2x⁴ + x3 + x + 5 = (x+1)(x⁴ -3x³+4x²-4x+5)

 

Now we need to further factor the new polynomial.

Our next possible factor we will try is x = 1...

If x=1 is a root then x-1 will be a factor.

 

1 | 1    -3    4    -4    5

   |        1   -2     2   -2

   ------------------------

     1    -2    2    -2    3

x=1 is NOT a root since there is a remainder of 3

 

Let's try x=5 as a root to see if (x-5) is a factor

 

5 | 1    -3    4    -4    5

   |        5   10   70  330

    -------------------------

     1     2    14   66  335

x=5 is NOT a root

 

Let's try x=-5 to see if (x+5) is a factor

 

-5 | 1    -3    4     -4        5

    |       -5  40  -220   1120

    ----------------------------

     1    -8    44  -224  1125

 

x=-5 is NOT a root either so x+5 is not a factor

 

Since the only real root we can find is x=1

the other roots must be complex or irrational

 

x⁵ - 2x⁴ + x³ + x + 5 = (x+1)(x⁴ -3x³+4x²-4x+5)

 

Answer 2

By trial and error, we see that -1 is a root.

 

So, by the Factor Theorem, x-(-1) = x+1 is a factor.  Divide synthetically by x+1 to get the other factor:

 

-1⌋ 1      -2      1       0      1     5

     _____-1___3____-4___4___-5

     1      -3      4       -4     5     0

 

So, the original polynomial is equivalent to (x+1)(x⁴-3x³+4x²-4x+5)

 

By the Rational Zero Theorem, The only possible rational roots of         x⁴-3x³+4x²-4x+5 are 1, -1, 5, and -5.  Checking, we find that none of them is a root.  So, use a graphing calculator to see if there are any irrational roots (x-intercepts other than -1).  If not, then all of the remaining roots are imaginary.

     

Answer 3

Hi Ramya,

 

Do you know how to do synthetic division?

 

Or maybe I should say, have you heard of synthetic division?