Use the Remainder Theorem and Factor Theorem to determine whether or not the first polynomial is a factor of the second.
1. What is the remainder of (-5x^3 + 14x + 12) ÷ (x - 2)
2. What is the remainder of (x^3 - x - 2) ÷ (x - 1)
3. (x - 1); (x^3 - x - 2) Is the first polynomial a factor of the second?
4. What is the remainder of (2x^3 - 9x^2 - 20) ÷ (x - 3)
5. (x - 3); (2x^3 - 9x^2 - 20) Is the first polynomial a factor of the second?
6. What is the remainder of (2x^3 + x^2 - 7x - 6) ÷ (x + 1)
7. (x + 1); (2x^3 + x^2 - 7x - 6) Is the first polynomial a factor of the second?
8. What is the remainder of (-5x^3 + 14x + 12) ÷ (x - 2)
9. (x - 2); (-5x^3 + 14x + 12) Is the first polynomial a factor of the second?
10. What is the remainder of (x^2 + 2x + 5) ÷ (x - 1)
1. What is the remainder of (-5x^3 + 14x + 12) ÷ (x - 2)
2. What is the remainder of (x^3 - x - 2) ÷ (x - 1)
3. (x - 1); (x^3 - x - 2) Is the first polynomial a factor of the second?
4. What is the remainder of (2x^3 - 9x^2 - 20) ÷ (x - 3)
5. (x - 3); (2x^3 - 9x^2 - 20) Is the first polynomial a factor of the second?
6. What is the remainder of (2x^3 + x^2 - 7x - 6) ÷ (x + 1)
7. (x + 1); (2x^3 + x^2 - 7x - 6) Is the first polynomial a factor of the second?
8. What is the remainder of (-5x^3 + 14x + 12) ÷ (x - 2)
9. (x - 2); (-5x^3 + 14x + 12) Is the first polynomial a factor of the second?
10. What is the remainder of (x^2 + 2x + 5) ÷ (x - 1)
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Answer:
To determine the remainders and whether the first polynomial is a factor of the second using the Remainder Theorem and Factor Theorem, we can apply polynomial division. Here are the solutions for each question:
1. Using polynomial division:
(-5x^3 + 14x + 12) ÷ (x - 2)
The division gives us a quotient of -5x^2 - 8x - 4 and a remainder of 4.
Therefore, the remainder is 4.
2. Using polynomial division:
(x^3 - x - 2) ÷ (x - 1)
The division gives us a quotient of x^2 + x + 2 and a remainder of 0.
Therefore, the remainder is 0.
3. (x - 1); (x^3 - x - 2)
To check if (x - 1) is a factor of (x^3 - x - 2), we can evaluate (x^3 - x - 2) when x = 1:
(1^3 - 1 - 2) = -2
Since the result is not zero, (x - 1) is not a factor of (x^3 - x - 2).
4. Using polynomial division:
(2x^3 - 9x^2 - 20) ÷ (x - 3)
The division gives us a quotient of 2x^2 - 3x + 6 and a remainder of -2.
Therefore, the remainder is -2.
5. (x - 3); (2x^3 - 9x^2 - 20)
To check if (x - 3) is a factor of (2x^3 - 9x^2 - 20), we can evaluate (2x^3 - 9x^2 - 20) when x = 3:
(2(3)^3 - 9(3)^2 - 20) = 0
Since the result is zero, (x - 3) is a factor of (2x^3 - 9x^2 - 20).
6. Using polynomial division:
(2x^3 + x^2 - 7x - 6) ÷ (x + 1)
The division gives us a quotient of 2x^2 - x - 8 and a remainder of 2.
Therefore, the remainder is 2.
7. (x + 1); (2x^3 + x^2 - 7x - 6)
To check if (x + 1) is a factor of (2x^3 + x^2 - 7x - 6), we can evaluate (2x^3 + x^2 - 7x - 6) when x = -1:
(2(-1)^3 + (-1)^2 - 7(-1) - 6) = 0
Since the result is zero, (x + 1) is a factor of (2x^3 + x^2 - 7x - 6).
8. Using polynomial division:
(-5x^3 + 14x + 12) ÷ (x - 2)
The division gives us a quotient of -5x^2 - 8x - 4 and a remainder of 4.
Therefore, the remainder is 4.
9. (x - 2); (-5x^3 + 14x + 12)
To check if (x - 2) is a factor of (-5x^3 + 14x + 12), we can evaluate (-5x^3 + 14x + 12) when x = 2:
(-5(2)^3 + 14(2) + 12) = 0
Since the result is zero, (x - 2) is a factor of (-5x^3 + 14x + 12).
10. Using polynomial division:
(x^2 + 2x + 5) ÷ (x - 1)
The division gives us a quotient of x + 3 and a remainder of 8.
Therefore, the remainder is 8.
In summary:
1. Remainder: 4
2. Remainder: 0
3. (x - 1) is not a factor of (x^3 - x - 2).
4. Remainder: -2
5. (x - 3) is a factor of (2x^3 - 9x^2 - 20).
6. Remainder: 2
7. (x + 1) is a factor of (2x^3 + x^2 - 7x -