Two polynomials P and D are given. Use either synthetic or long division to divide
P(x)
by
D(x),
and express P in the form
P(x) = D(x) · Q(x) + R(x).
P(x) = −x3 − 4x + 3, D(x) = x + 1
so do the division. You can find videos on either method, which are hard to show here, where text formatting is impossible.
You should find that
-x^3 - 4x + 3 = (x+1)(-x^2+x-5) + 8
To divide P(x) = -x^3 - 4x + 3 by D(x) = x + 1 using long division, follow these steps:
Step 1: Arrange the polynomials in descending powers of x and make sure that all powers are present. If a power is missing, include it with a coefficient of 0. Your arranged polynomials should look like:
-x^3 + 0x^2 - 4x + 3 | x + 1
Step 2: Divide the leading term of the dividend (P(x)) by the leading term of the divisor (D(x)). In this case, divide -x^3 by x:
-x^3 ÷ x = -x^2
Step 3: Multiply the result from Step 2 by the divisor (D(x)) and write it below the dividend (P(x)):
-x^2
-x^3 + 0x^2 - 4x + 3 | x + 1
-x^3 + 0x^2
Step 4: Subtract the result obtained in Step 3 from the dividend and bring down the next term (-4x):
-x^2
-x^3 + 0x^2 - 4x + 3 | x + 1
-x^3 + 0x^2
_________
-4x
Step 5: Repeat Steps 2-4 until the degree of the remaining polynomial is less than the degree of the divisor.
Step 2 (repeated): Divide the leading term of the new dividend (-4x) by the leading term of the divisor (x):
-4x ÷ x = -4
Step 3 (repeated): Multiply the result from Step 2 by the divisor (D(x)) and write it below the new dividend:
-x^2 - 4
-x^3 + 0x^2 - 4x + 3 | x + 1
-x^3 + 0x^2
_________
-4x
Step 4 (repeated): Subtract the result obtained in Step 3 from the new dividend and bring down the next term (3):
-x^2 - 4
-x^3 + 0x^2 - 4x + 3 | x + 1
-x^3 + 0x^2
_________
-4x + 3
Step 5 (repeated): The degree of the remaining polynomial (-4x + 3) is less than the degree of the divisor (x + 1), so the division process is complete.
Step 6: Write the result in the form P(x) = D(x) · Q(x) + R(x):
P(x) = (x^2 - 4) · (x + 1) + (-4x + 3)
Therefore, P(x) = (x^2 - 4)(x + 1) + (-4x + 3).
To divide P(x) by D(x), which are given as P(x) = -x^3 - 4x + 3 and D(x) = x + 1 respectively, we can use either synthetic division or long division method. I'll explain both methods, and you can choose the one you prefer.
1. Synthetic Division:
- Write down the coefficients of P(x) in descending order based on the powers of x: -1, 0, -4, and 3.
- Arrange the coefficients in a line and place the divisor, x + 1, to the left of them.
-1 0 -4 3
x + 1
- Bring down the first coefficient, which is -1, and write it below the line:
-1 0 -4 3
x + 1
-1
- Multiply -1 by the divisor, x + 1, and write the result below the next coefficient:
-1 0 -4 3
x + 1
-1
-1
- Add the multiplied result with the next coefficient, 0:
-1 0 -4 3
x + 1
-1
-1
--------
-1
- Repeat the previous steps for the new result, -1:
-1 0 -4 3
x + 1
-1 -1
--------
-1 -1
- Continue until you have used all the coefficients. The final result will be the remainder of the division.
Therefore, using synthetic division, we have P(x) = (x + 1)(-x^2 - x - 1) - 1 as the quotient, and -1 as the remainder. So, P(x) = (x + 1)(-x^2 - x - 1) - 1.
2. Long Division:
- Write down P(x) and D(x) in long division format:
-x^2 - 5x + 1
x + 1 | -x^3 - 4x + 3
- Divide the first term of P(x) by the first term of D(x), -x^3 / x = -x^2.
- Multiply the divisor, x + 1, by -x^2 and write the result below P(x):
-x^2 - 5x + 1
x + 1 | -x^3 - 4x + 3
-x^3 - x^2
- Subtract the result from P(x):
-x^2 - 5x + 1
x + 1 | -x^3 - 4x + 3
- (-x^3 - x^2)
----------------
-4x^2 + 3
- Bring down the next term, -4x:
-x^2 - 5x + 1
x + 1 | -x^3 - 4x + 3
- (-x^3 - x^2)
-4x^2 + 3
- Divide the first term of the new polynomial by the first term of D(x), -4x^2 / x = -4x.
- Multiply the divisor, x + 1, by -4x and write the result below the previous result:
-x^2 - 5x + 1
x + 1 | -x^3 - 4x + 3
- (-x^3 - x^2)
-4x^2 + 3
- (-4x^2 - 4x)
----------------
7x + 3
- Subtract the result from the polynomial:
-x^2 - 5x + 1
x + 1 | -x^3 - 4x + 3
- (-x^3 - x^2)
-4x^2 + 3
- (-4x^2 - 4x)
7x + 3
- (7x + 7)
-----------------------
-4
- The result is -4, which is the remainder of the division.
Therefore, using long division, we have P(x) = (x + 1)(-x^2 - 5x + 7x + 3) - 4 as the quotient, and -4 as the remainder. So, P(x) = (x + 1)(-x^2 + 2x + 3) - 4.
Both methods yield the same result, so you can choose either expression for P(x) in the form P(x) = D(x) · Q(x) + R(x).