Question

solve for x using the factor theorem and long division. a. 2x^3-3x^2-5x+6=0   b. 8x^3+4x^2-18x-9=0   c. 2x^3+10x^2+13x+5=0   d. -2x^3-12x^2+x+60=0

Answer 1

From Rational Zeroes Theorem, the rational zeroes of the polynomial are of the form where p is factor of constant term and q is factor of the leadin coefficient. For the given equation, factors of constant 6 are 1,2,3,6 and factors of leading coefficinet 2 are 1, 2

Therefore, rational zeroes are of the form

Find Remainder Theorem, if f(a)=0 implies a=root of f(x)

Therefore, x=1 is root of f(x) implies (x-1) is a factor of f(x)

Divide f(x) by (x-1) by Long Division

Therefore, f(x) can be written as

Factorize the quadratic term

That is

Therefore,