Math question?

Question

a) Show that 1 is a zero of the polynomial P(x)=x^4-2x^3+7x-6

b) If α,β,and γ are the other 3 zeros , find the value of α+β+ γ and αβγ

Answer 1

• a).

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.

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.

The polynomial is p (x ) = x⁴ - 2x³ + 7x - 6.

If p/q is a rational zero, then p is a factor of 6 and q is a factor of 1.

The possible values of p are   ± 1,  ± 2, and ± 3.

The possible values for q are ± 1.

By the Rational Roots Theorem, the only possible rational roots are, p /q = ± 1, ± 2, and ± 3.

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

p /q	1	-  2	0	7	- 6
- 1	1	- 3	3	4	- 10
1	1	- 1	- 1	6	0

Since, p (1) = 0, x = 1 is a zero.

Answer 2

Contd.....

• b).

If α, β,  γ, and δ are the roots of the polynomial is ax⁴ + bx³ + cx² + dx + e, then,

1. Sum of the roots(α + β + γ + δ) = - coefficient of x³/coefficient of x⁴
2. Product of the roots(αβγδ) = - constant term/coefficient of x⁴.

The one root of the  polynomial : x⁴ - 2x³ + 7x - 6 is 1 and assume that other three roots are α, β,  γ.

• Sum of the roots(α + β + γ + 1) = - coefficient of x³/coefficient of x⁴ = - (- 2)/1 = 2.

α + β + γ + 1 = 2

⇒ α + β + γ = 2 - 1 = 1.

• Product of the roots(α * β * γ * 1) = - constant term/coefficient of x⁴ = - (- 6)/1 = 6

⇒ αβγ = 6.