P(x)= 2x^4+5x^3-22x^2-25x+60?

Question

Update : a. Use synthetic division to show that (3/2) is a zero of P. 
b. Use your answer to part a, more synthetic division, and the square root property to find the other zeros of P. 
c. Use your answer from part b to write the completely factored form of P(x).

Answer 1

a)The polynomial P(x) = 2x⁴ +  5x³ - 22x²  - 25x + 60

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

If p/q is a rational zero, then p is a factor of 60 and q is a factor of 2.

The possible values of p are   ± 1, ± 2, ± 3, ± 4, ± 5, ± 6, ± 10, ± 15, ± 20, ± 30 and ± 60.

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

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 3,± 4, ± 5, ± 6, ± 10, ± 15, ± 20, ± 30 , ± 60, ± 1/2, ± 3/2, ± 5/2 and ± 15/2.

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

p/q	2	5	- 22	- 25	60
1	2	7	- 15	- 40	20
2	2	9	- 4	- 33	- 6
3	2	11	11	8	84
3/2	2	8	- 10	- 40	0

Since P(3/2) = 0,  x = 3/2 is a zero of P(x).

Answer 2

b) Observe the solution a,

Since P(3/2) = 0,  x = 3/2 is a zero. The depressed polynomial is  2x³ + 8x² - 10x - 40 = 0.

If p/q is a rational zero, then p is a factor of 40 and q is a factor of 2.

The possible values of p are   ± 1, ± 2, ± 4, ± 5, ± 8, ± 10, ± 20 and ± 40.

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

By the Rational Roots Theorem, the only possible rational roots are, p/q =± 1, ± 2, ± 4, ± 5, ± 8, ± 10, ± 20, ± 40, ± 1/2 and ± 5/2.

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

p/q	2	8	- 10	- 40
1	2	10	0	- 40
2	2	12	14	- 12
- 4	2	0	-10	0

Since P(- 4) = 0,  x = - 4 is a zero. The depressed polynomial is  2x² - 10

Solve the related equation 2x² - 10 = 0.

2x² = 10

x² = 5

Apply square root on each side.

x = ± √5

Zeros of P(x) x = -4 , 3/2 , √5 and - √5.

Answer 3

c)Observe the solution b,

From Factor theorem,

When x - c is a factor of the polynomial then f(c) = 0.

Real zeros are -4 , 3/2 , √5 and - √5

Factor form of 2x⁴ +  5x³ - 22x²  - 25x + 60 = (x + 4)(2x - 3)(x + √5)(x - √5).