Remainder Theorem and Factor Theorem

What does the Remainder Theorem state?

• A)The remainder when a polynomial ( p(x) ) is divided by

  x−a

  is ( p(a) ).

• B) The remainder when a polynomial ( p(x) ) is divided by

  x−a

  is ( p(0) ).

• C) The remainder when a polynomial ( p(x) ) is divided by

  x−a

  is ( a ).

• D) The remainder is always zero.

What is the remainder when p(x)=x³−2x²+3x−4 is divided by x−1 ?

• A) -2

• B) -1

• C) 0

• D) 1

According to the Factor Theorem, when is x−a a factor of a polynomial ( p(x) )?

• A) If p(a)=1

• B) If p(a)=0

• C) If p(0)=a

• D) If p(x)=a

Is x+2 a factor of f(x) = x⁴+x³−5x²−x+6 ?

• A) Yes, because

  f(−2)=0

• B) No, because

  f(−2)≠0

• C) Yes, because

  f(2)=0

• D) No, because

  f(2)≠0

What is the value of ( k ) such that x−3 is a factor of g(x)=x³−4x²+kx−6 ?

• A) 4

• B) 5

• C) 6

• D) 7

A polynomial ( p(x) ) leaves a remainder of 5 when divided by x−2 and -3 when divided by

x+1 . What is the remainder when divided by x²−x−2 ?

• A) x+1

• B) 2x+1

• C) x−3

• D) 4x−3

What is the value of ( m ) such that x+1 is a factor of h(x)=x³+mx²−4x+5 ?

• A) -5

• B) -4

• C) -3

• D) -2

According to the Remainder Theorem, what is the remainder when a polynomial ( p(x) ) is divided by x−c?

• C) c

• D) 0

• A) p(c)

• B) p(0)

If a polynomial ( p(x) ) has a factor of x+3, what can be concluded about p(−3)?

• C) p(−3) must be −3

• B) p(−3) must be 1

• A) p(−3) must be 0

• D) p(−3) can be any value

Given x−2 and x+3

are factors of h(x)=x³+ax²+bx−12 , what are the values of ( a ) and ( b )?

• A) a=−2,b=−3

• B) a=2,b=−3

• C) a=−2,b=3

• D) a=2,b=3