How i do the maths 2x^4-9x^3+14x^2-9x+2=0?

Question

please show steps

Answer 1

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

 2x ⁴- 9x ³ + 14x ²- 9x  + 2 = 0

p = 2, q  = 2

If p /q  is a rational zero, then p  is a factor of 2 and q  is a factor of 2.

The possible values of p  are   ± 1 and  ± 2.

The possible values for q  are ± 1, ± 2.

So, p /q  = ± 1, ± 2.

Make a table for the synthetic division and test possible  zeros.

p /q	2	-9	14	-9	2
1	2	-7	7	-2	0

Since f(1)   =   0,   x   =   1 is a zero. The depressed polynomial is 2x³  - 7x² + 7x - 2 = 0.

2x³  - 7x² + 7x - 2 = 0.

 If p /q  is a rational zero, then p is a factor of 2 and q is a factor of 2.

The possible values of p  are   ± 2.

The possible values for q  are ± 1 and ± 2.

So, p /q  = ± 1,   ± 2

Make a table for the synthetic division and test possible real zeros.

p /q	1	-7	7	-2
1	2	-5	2	0

Since the depressed polynomial of this zero, 2x²- 5x + 2, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation.

2x²- 5x + 2 = 0

2x²- 4x - x + 2 = 0

2x (x - 2) - 1(x -2) = 0

(x - 2) (2x -1) = 0

x - 2 = 0 and 2x - 1 = 0

x  = 2 and 2x  = 1

x  = 2 and x = 1/2

2x ⁴- 9x ³+14x ²- 9x + 2 = (x -1)²(x - 2)(2x - 1)

Roots of the polynomial x = 1, 2 and 1/2.