Factors of polynomials

Question 1

For polynomial f(x), what if f(c)=0?

Answer 1

Then x-c is a factor of f(x)

Question 2

For polynomial f(x), what if f(b/a)=0?

Answer 2

Then ax-b is a factor of f(x)

Question 3

What if f(x) is divided by x-c?

Answer 3

The remainder is f(c)

Question 4

What if f(x) is divided by ax-b?

Answer 4

The remainder is f(b/a)

Question 5

What are the possible rational roots for polynomial f(x)=a_nxⁿ+a_(n-1)x^(n-1)+…+a₁x¹+a₀?

Answer 5

Factors of a₀/factors of a_n

Question 6

If x-5 is a factor of f(x) = 2x³-x²-41x-20 solve f(x) = 0

Answer 6

2x³-x²-41x-20 = (x-5)(ax²+bx+c) the coefficient of x³ is 2 therefore a = 2 and the coefficient of x⁰ is -20 therefore -5c = -20
2x³-x²-41x-20 = (x-5)(2x²+bx+4)
2x³-x²-41x-20 = 2x^{3</sup+(b-10)x2+(4-5b)x-20
b-10 = 1
b = 9}