Perform the following polynomials divisions: (12 marks)
(x2 - 7x - 12) ÷ (x + 2)
(x3 + x2 – 3x + 1) ÷ (x – 1)
(3x3 + 5x2 – 3x – 5) ÷ (x + 1)
(x4 - 3x3 + 3x + 2) ÷ (x - 2)
Based on the results of question 1, what do you notice when the divisor is a factor of the polynomial? (1 mark)
correction
(x^2 - 7x - 12) ÷ (x + 2)
(x^3 + x^2 – 3x + 1) ÷ (x – 1)
(3x^3 + 5x^2 – 3x – 5) ÷ (x + 1)
(x^4 - 3x^3 + 3x + 2) ÷ (x - 2)
(x^2 - 7x - 12) ÷ (x + 2) = (x-9) remainder 6
(x^3 + x^2 – 3x + 1) ÷ (x – 1) = (x^2+2x-1) remainder 0
(3x^3 + 5x^2 – 3x – 5) ÷ (x + 1) = 3x^2+2x-5 remainder 0
(x^4 - 3x^3 + 3x + 2) ÷ (x - 2) = x^3-x^2-2x-1 remainder 0
To perform polynomial division, you need to follow these steps:
Step 1: Arrange the polynomials in descending order of degree.
Step 2: Divide the first term of the dividend (numerator) by the first term of the divisor (denominator). This will be the first term of the quotient.
Step 3: Multiply the entire divisor by the first term of the quotient and subtract the result from the dividend.
Step 4: Repeat steps 2 and 3 until you have divided all the terms of the dividend.
Step 5: The terms left after division will be the remainder. The quotient is the sum of the terms in the quotient column.
Now let's perform polynomial division for the given examples.
1. (x^2 - 7x - 12) ÷ (x + 2)
Step 1: The dividend is in descending order, so no rearrangement is required.
Step 2: First term of the quotient = (x^2 ÷ x) = x.
Step 3: Multiply (x + 2) by x and subtract the result from the dividend:
(x + 2) * x = x^2 + 2x
(x^2 - 7x - 12) - (x^2 + 2x) = -9x - 12
Step 4: Repeat steps 2 and 3 with the new dividend.
Next term of the quotient = (-9x ÷ x) = -9.
Multiply (x + 2) by -9 and subtract the result from the new dividend:
(x + 2) * -9 = -9x - 18
(-9x - 12) - (-9x - 18) = 6
Step 4: We can't divide 6 by (x + 2) anymore, so the division is complete.
The quotient is (x - 9) with a remainder of 6.
2. (x^3 + x^2 – 3x + 1) ÷ (x – 1)
Step 1: The dividend is already in descending order.
Step 2: First term of the quotient = (x^3 ÷ x) = x^2.
Step 3: Multiply (x - 1) by x^2 and subtract the result from the dividend:
(x - 1) * x^2 = x^3 - x^2
(x^3 + x^2 – 3x + 1) - (x^3 - x^2) = 2x^2 - 3x + 1
Step 4: Repeat steps 2 and 3 with the new dividend.
Next term of the quotient = (2x^2 ÷ x) = 2x.
Multiply (x - 1) by 2x and subtract the result from the new dividend:
(x - 1) * 2x = 2x^2 - 2x
(2x^2 - 3x + 1) - (2x^2 - 2x) = -x + 1
Step 4: We can't divide -x + 1 by (x - 1) anymore, so the division is complete.
The quotient is (x^2 + 2x) with a remainder of (-x + 1).
3. (3x^3 + 5x^2 – 3x – 5) ÷ (x + 1)
Following the steps, we find that the quotient is (3x^2 - 2x + 1) with a remainder of (-6).
4. (x^4 - 3x^3 + 3x + 2) ÷ (x - 2)
The quotient is (x^3 - x^2 - 2x - 4) with no remainder.
Based on these results, we notice that when the divisor is a factor of the polynomial, the remainder is zero. In other words, there is no remainder, and the polynomial is evenly divisible by the divisor.