find the integral zero of 4x^3=20x^2-X-5 - Maths - Polynomials

Question

find the integral zero of
4x^3=20x^2-X-5

Koka Sri Lakshmi Divya Sai · Accepted Answer

The equation is 4x^3-20x^2-x+5

Integral Zero Theorem says that zeroes and roots of this polynomial,must be the factors of the constant term,this means the

possible real roots of this polynomial are 1,-1,5,-5

Substituting values for P(x),we get

${P (x)=4x}^{3} - 20 x^{2} - x + 5 P (1) = 4 (1)^{3} - 20 (1)^{2} - 1 + 5 = 4 - 20 - 1 + 5 = - 12 P (- 1) = 4 (- 1)^{3} - 20 (- 1)^{2} + 1 + 5 = - 4 - 20 + 1 + 5 = - 18 P (5) = 4 (5)^{3} - 20 (5)^{2} - 5 + 5 = 4 (125) - 20 (25) - 5 + 5 = 500 - 500 - 5 + 5 = 0 ∴ 5 is the root of the given equation Now we can divide the polynomial by this factor to find the other factors$

$Now we have 4 x^{3} - 20 x^{2} - x + 5 = (x - 5) (4 x^{2} - 1) The original polynomial factors as \Rightarrow 4 x^{3} - 20 x^{2} - x + 5 = (x - 5) (2 x - 1) (2 x + 1)$