Remainder Theorem (Grades 11-12)

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Remainder Theorem

1. 
What is the remainder of [math]3x^2+x-6[/math] divided by [math]x-1[/math]?
  1. [math]-2[/math]
  2. [math]x-4[/math]
  3. [math]-4[/math]
  4. [math]-6[/math]
2. 
Which of the following are factors of the polynomial [math]p(x) = 2x^4-x^3-17x^2+25x-6 ? [/math] There may be more than one answer.
  1. [math]2x-3[/math]
  2. [math]x-5[/math]
  3. [math]3x-4[/math]
  4. [math]x-2[/math]
3. 
For polynomial [math]f(x)[/math], if [math]f(0)=0[/math], which of the following MUST be a factor of [math]f(x) ?[/math]
  1. [math]x[/math]
  2. [math]x-4[/math]
  3. [math]x-1[/math]
  4. [math]x+1[/math]
4. 
Given polynomial [math]p(x)[/math], such that [math]p(3) = 23[/math], what is a possible answer for [math]p(x) [/math] divided by [math]x-3 ?[/math]
  1. [math]3x + 3 + 2/(x-3)[/math]
  2. [math]4x^2 - 3x + 1 + 23/(x-3)[/math]
  3. [math]23x^2 + x[/math]
  4. [math]x + 23[/math]
5. 
Given polynomial [math]q(x)[/math], such that [math]q(4)=0[/math], what is a possible answer for [math]q(x)[/math] divided by [math]x-4 ?[/math]
  1. [math]3x^2 + 3 - 4/(x-4)[/math]
  2. [math]x-10[/math]
  3. [math]x^2 + 1 + 2/(x-4)[/math]
  4. [math]x + 2 - 4/(x-4)[/math]
6. 
For polynomials [math]p(x)[/math] and [math]q(x)[/math], where [math](p(x))/(x-1) = q(x) + b/(x-1)[/math], what value(s) of [math]b[/math] ensure that [math]p(1) = 0 ?[/math] There may be more than one correct answer.
  1. [math]x-1[/math]
  2. [math]1[/math]
  3. [math]0[/math]
  4. Any real number
7. 
For polynomials [math]f(x)[/math] and [math]g(x)[/math], where [math](f(x))/(x-3) = g(x) + c/(x-3)[/math], what value(s) of c ensure that [math]f(3)!=0 ?[/math] There may be more than one correct answer.
  1. 3
  2. 0
  3. 1
  4. -3
8. 
Given that [math]x+3[/math] is factor of polynomial [math]p(x) = ax^3+53x^2+46x-15[/math], what is the value of [math]a ?[/math]
  1. [math]200/9[/math]
  2. [math]-3[/math]
  3. [math]"Not enough information"[/math]
  4. [math]12[/math]
9. 
If [math]x-2[/math] is a factor of polynomial [math]p(x) = 8x^4 + 6x^3 - ax^2 - 13x + 30[/math], what is the value of [math]a ?[/math]
  1. Not enough information
  2. 34
  3. 45
  4. 180
10. 
Given polynomial [math]p(x) = ax^3 - 5x^2 + bx + 4[/math], where [math]x-2[/math] is a factor of [math]p(x)[/math], and where [math](p(x))/(x+3) = q(x) - 110/(x+3)[/math], [math]q(x)[/math] being the quotient of [math]p(x)[/math] divided by [math]x+3[/math], what are the values of [math]a[/math] and [math]b ?[/math]
  1. [math]a = 2, b=-3[/math]
  2. [math]a=-15, b=68[/math]
  3. [math]a=3, b=-4[/math]
  4. [math]a=-31/5, b=164/5[/math]

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