Questions

1. Question 1

   Find remainder and quotient by simplifying the following:

   (i)

   $(3x^2 - x + 2) \div (x-1)$

   (ii)

   $(x^3 + 12x^2 - 3x + 4) \div (x-2)$

   (iii)

   $(x^4 - 5x^3 - 8x^2 + 13x + 12) \div (x-6)$

   (iv)

   $(5x^4 - 3x^3 + 2x^2 - 1) \div (x^2 + 4)$

   (v)

   $(3x^4 - 5x^3 + 4x - 6) \div (x^2 - 3x + 5)$

   SolutionTheory
2. Question 2

   Use the remainder theorem to find the remainder when the first polynomial is divided by the second polynomial.

   (i)

   $x^2 + 5x + 6$, $x-2$

   (ii)

   $x^3 + 5x^2 + 6$, $x+1$

   (iii)

   $x^4 + x^3 + x^2 + x + 1$, $x-1$

   (iv)

   $x^4 + x^2 + 1$, $x+3$

   (v)

   $x^4 + x^3 + 2$, $x+2$

   SolutionTheory
3. Question 3

   Use the factor theorem to determine if the first polynomial is a factor of the second polynomial.

   (i)

   $x+1$, $x^2 - 1$

   (ii)

   $x-2$, $x^2 - 5x + 6$

   (iii)

   $x+1$, $x^3 + x^2 + x - 3$

   (iv)

   $x-2$, $x^3 + x^2 - 7x + 2$

   (v)

   $x-3$, $x^4 - 3x^3 + x^2 - x + 1$

   SolutionTheory
4. Question 4

   Use synthetic division to show that $x$ is the zero of the polynomial and use the result to factorize the polynomial completely.

   (i)

   $x^3 - 7x + 6$, $x=2$

   (ii)

   $x^3 - 28x - 48$, $x=-4$

   (iii)

   $2x^4 + 7x^3 - 4x^2 - 27x - 18$, $x=2, x=-3$

   SolutionTheory
5. Question 5

   Use synthetic division to find the quotient and the remainder when the polynomial $x^4 - 10x^2 - 2x + 4$ is divided by $x+3$.

   SolutionTheory
6. Question 6

   If $x+1$ and $x-2$ are factors of $x^3 - px^2 + qx + 2$. Using synthetic division, find the values of $p$ and $q$.

   SolutionTheory
7. Question 7

   When the polynomial $4x^4 + 2x^3 + kx^2 + 13$ is divided by $x+1$, the remainder is 16. Find the value of $k$.

   SolutionTheory
8. Question 8

   When the polynomial $x^3 + x^2 + x + k$ is divided by $x+1$, the remainder is 7. Find the value of $k$.

   SolutionTheory
9. Question 9

   Use factor theorem to find the values of $p$ and $q$ if $x+1$ and $x-2$ are the factors of the polynomial $x^3 + px^2 + qx + 3$.

   SolutionTheory
10. Question 10

    Use factor theorem to find the values of $a$ and $b$ if $-2$ and 2 are the roots of the polynomial $2x^3 + 4x^2 + ax + b$.

    SolutionTheory