Applying the Remainder Theorem - Algebra 2

1. Find the value of $k$ so that $x-2$ is a factor of $p(x)=x^2+kx-6$.

   $k=1$. Set $p(2)=0$: $4+2k-6=0$, so $k=1$.

2. What is the remainder when dividing $p(x)=x^4-16$ by $x+2$?

   $p(-2)=0$. Calculate $p(-2)=(-2)^4-16=16-16=0$.

3. What is the remainder when dividing $p(x)=3x^2-12x+9$ by $x-1$?

   $p(1)=0$. Calculate $p(1)=3(1)^2-12(1)+9=3-12+9=0$.

4. What is the remainder when dividing $p(x)=x^2-1$ by $x-\frac{1}{3}$?

   $p\left(\frac{1}{3}\right)=-\frac{8}{9}$. Calculate $p\left(\frac{1}{3}\right)=\left(\frac{1}{3}\right)^2-1=\frac{1}{9}-1=-\frac{8}{9}$.

5. Identify the remainder when dividing $p(x)=x^2+x+1$ by $x-1$.

   $p(1)=3$. Calculate $p(1)=1^2+1+1=1+1+1=3$.

6. If $p(0)=5$, what is the remainder when dividing $p(x)$ by $x$?

   Remainder $=5$. Dividing by $x$ means evaluating at $x=0$.

7. What is the remainder when dividing $p(x)=x^3-2x^2+x-2$ by $x-2$?

   $p(2)=0$. Calculate $p(2)=8-8+2-2=0$.

8. If the remainder on division by $x-a$ is $r$, what is $p(a)$ equal to?

   $p(a)=r$. The remainder equals the polynomial evaluated at $a$.

9. Identify the remainder when dividing $p(x)=5x^2+2x-3$ by $x-0$.

   $p(0)=-3$. Calculate $p(0)=5(0)^2+2(0)-3=-3$.

10. What is the remainder when dividing $p(x)=x^3-9x$ by $x+3$?

    $p(-3)=0$. Calculate $p(-3)=(-3)^3-9(-3)=-27+27=0$.

11. What is the remainder when dividing $p(x)=4x^3+1$ by $x-\frac{1}{2}$?

    $p\left(\frac{1}{2}\right)=\frac{3}{2}$. Calculate $p\left(\frac{1}{2}\right)=4\left(\frac{1}{2}\right)^3+1=\frac{1}{2}+1=\frac{3}{2}$.

12. If dividing $p(x)$ by $x-9$ gives remainder $7$, what is the value of $p(9)$?

    $p(9)=7$. By the Remainder Theorem, $p(a)$ equals the remainder.

13. If $p(6)=-3$, what is the remainder when dividing $p(x)$ by $x-6$?

    Remainder $=-3$. By the Remainder Theorem, the remainder equals $p(6)$.

14. Identify the value of $a$ used in the Remainder Theorem when the divisor is $x+7$.

    $a=-7$. For divisor $x+7=x-(-7)$, the value $a=-7$.

15. Which condition guarantees that $(x-a)$ is a factor of $p(x)$: $p(a)=0$ or $p(a)\neq 0$?

    $p(a)=0$. When $p(a)=0$, the remainder is zero, making $(x-a)$ a factor.

16. If dividing $p(x)$ by $x+2$ gives remainder $0$, what is the value of $p(-2)$?

    $p(-2)=0$. If remainder is 0, then $p(a)=0$ by the theorem.

17. What is the remainder when dividing $p(x)=x^2-2x+2$ by $x+1$?

    $p(-1)=5$. Calculate $p(-1)=(-1)^2-2(-1)+2=1+2+2=5$.

18. What does the Factor Theorem state using $p(a)$ and the factor $(x-a)$?

    $p(a)=0\iff (x-a)$ is a factor of $p(x)$. The Factor Theorem combines remainder and factorization.

19. Find the remainder when dividing $p(x)=x^3+1$ by $x-1$.

    $p(1)=2$. Calculate $p(1)=1^3+1=1+1=2$.

20. Which divisor corresponds to evaluating $p(-5)$ by the Remainder Theorem: $x-5$ or $x+5$?

    $x+5$. To evaluate $p(-5)$, divide by $x-(-5)=x+5$.

21. Which divisor corresponds to evaluating $p(5)$ by the Remainder Theorem: $x-5$ or $x+5$?

    $x-5$. To evaluate $p(5)$, divide by $x-5$.

22. If $p(-4)=12$, what is the remainder when dividing $p(x)$ by $x+4$?

    Remainder $=12$. By the Remainder Theorem, the remainder equals $p(-4)$.

23. Which statement is true if the remainder on division by $x-4$ is $0$?

    $(x-4)$ is a factor of $p(x)$. A zero remainder means the divisor is a factor.

24. If $(x-a)$ is a factor of $p(x)$, what must the remainder be when dividing by $(x-a)$?

    Remainder $=0$. If $(x-a)$ is a factor, then $p(a)=0$.

25. Identify the remainder when dividing $p(x)=x^2+x+1$ by $x+1$.

    $p(-1)=1$. Calculate $p(-1)=(-1)^2+(-1)+1=1-1+1=1$.

26. What is the remainder when dividing $p(x)$ by $x+5$ in terms of $p$?

    Remainder $=p(-5)$. Since $x+5=x-(-5)$, we have $a=-5$.

27. Identify the value of $a$ used in the Remainder Theorem when the divisor is $x-7$.

    $a=7$. For divisor $x-7$, the value $a=7$.

28. Find the value of $k$ so that $x-1$ is a factor of $p(x)=x^3-4x^2+kx+3$.

    $k=0$. Set $p(1)=0$: $1-4+k+3=0$, so $k=0$.

29. What is the remainder when dividing $p(x)=x^4-16$ by $x-2$?

    $p(2)=0$. Calculate $p(2)=2^4-16=16-16=0$.

30. Find the remainder when dividing $p(x)=x^3-1$ by $x+1$.

    $p(-1)=-2$. Calculate $p(-1)=(-1)^3-1=-1-1=-2$.

31. If $p(-2)=0$, which linear factor must divide $p(x)$: $x-2$ or $x+2$?

    $x+2$. If $p(-2)=0$, then $(x+2)$ is a factor.

32. What is the remainder when dividing $p(x)=x^3-4x+1$ by $x-2$?

    $p(2)=1$. Calculate $p(2)=2^3-4(2)+1=8-8+1=1$.

33. What is the remainder when dividing $p(x)=x^3-2x^2+x-2$ by $x+1$?

    $p(-1)=-6$. Calculate $p(-1)=-1-2-1-2=-6$.

34. What is the remainder when dividing $p(x)=x^4+x^3-x-1$ by $x+1$?

    $p(-1)=0$. Calculate $p(-1)=(-1)^4+(-1)^3-(-1)-1=1-1+1-1=0$.

35. What is the remainder when dividing $p(x)=2x^3-x^2+5$ by $x-1$?

    $p(1)=6$. Calculate $p(1)=2(1)^3-(1)^2+5=2-1+5=6$.

36. What is the remainder when dividing $p(x)=x^3-9x$ by $x-3$?

    $p(3)=0$. Calculate $p(3)=3^3-9(3)=27-27=0$.

37. If $p(2)=0$, which linear factor must divide $p(x)$: $x-2$ or $x+2$?

    $x-2$. If $p(2)=0$, then $(x-2)$ is a factor.

38. What is the remainder when dividing $p(x)=3x^2-12x+9$ by $x-3$?

    $p(3)=0$. Calculate $p(3)=3(3)^2-12(3)+9=27-36+9=0$.

39. Find the value of $k$ so that $x+3$ is a factor of $p(x)=2x^2+kx-9$.

    $k=3$. Set $p(-3)=0$: $18-3k-9=0$, so $k=3$.

40. Find the value of $k$ so that $x+2$ is a factor of $p(x)=x^3+kx^2-4x+8$.

    $k=0$. Set $p(-2)=0$: $-8+4k+8+8=0$, so $k=0$.

41. Find the remainder when dividing $p(x)=x^3+1$ by $x+1$.

    $p(-1)=0$. Calculate $p(-1)=(-1)^3+1=-1+1=0$.

42. What is the remainder when dividing $p(x)=x^4+x^3-x-1$ by $x-1$?

    $p(1)=0$. Calculate $p(1)=1^4+1^3-1-1=1+1-1-1=0$.

43. What does the Remainder Theorem state for the remainder when dividing $p(x)$ by $x-a$?

    Remainder $=p(a)$. This is the core statement of the Remainder Theorem.

44. What is the remainder when dividing $p(x)=x^2+3x-10$ by $x-2$?

    $p(2)=0$. Calculate $p(2)=2^2+3(2)-10=4+6-10=0$.

45. What is the remainder when dividing $p(x)=x^3+2x^2-x-2$ by $x+2$?

    $p(-2)=0$. Calculate $p(-2)=(-2)^3+2(-2)^2-(-2)-2=-8+8+2-2=0$.

46. What is the remainder when dividing $p(x)=2x^3-x^2+5$ by $x+1$?

    $p(-1)=2$. Calculate $p(-1)=2(-1)^3-(-1)^2+5=-2-1+5=2$.

47. What is the remainder when dividing $p(x)=x^2-2x+2$ by $x-1$?

    $p(1)=1$. Calculate $p(1)=1^2-2(1)+2=1-2+2=1$.

48. What is the remainder when dividing $p(x)$ by $x-3$ in terms of $p$?

    Remainder $=p(3)$. By the Remainder Theorem, substitute $a=3$.

49. Find the remainder when dividing $p(x)=x^3-1$ by $x-1$.

    $p(1)=0$. Calculate $p(1)=1^3-1=1-1=0$.

50. What is the remainder when dividing $p(x)=x^2+3x-10$ by $x+5$?

    $p(-5)=0$. Calculate $p(-5)=(-5)^2+3(-5)-10=25-15-10=0$.

51. What is the remainder when dividing $p(x)=x^3+2x^2-x-2$ by $x-1$?

    $p(1)=0$. Calculate $p(1)=1^3+2(1)^2-1-2=1+2-1-2=0$.

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