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).
a) P(x) = −x^3 − 5x + 3, D(x) = x + 1
b) P(x) = 27x^5 − 9x^4 + 3x^2 − 3, D(x) = 3x^2 − 3x + 1
it is pretty hard to show this in this format. Try:
www.wolframalpha.com
in format
( −x^3 − 5x + 3) / (x + 1)
−x^3 − 5x + 3, D(x) = x + 1
-1 0 -5...3
....1..-1..6
----------------
-1..1..-6..9
so that's (x+1)(-x^2+x-6) R 9
a) To find the quotient and remainder when dividing P(x) = -x^3 - 5x + 3 by D(x) = x + 1, we can use long division or synthetic division.
Long Division:
Step 1: Write the dividend (P(x)) and divisor (D(x)) in descending powers of x:
-x^3 - 5x + 3 ÷ x + 1
Step 2: Divide the first term of the dividend (-x^3) by the first term of the divisor (x) to get -x^2 as the first term of the quotient. Multiply this term by the entire divisor and subtract it from the dividend:
-x^3 - 5x + 3
-(x^3 + x^2)
-6x + 3
Step 3: Bring down the next term (-5x) and continue the division:
-6x + 3 ÷ x + 1
Step 4: Divide the first term of the new divisor (-6x) by the first term of the divisor (x) to get -6 as the next term of the quotient. Multiply this term by the entire divisor and subtract it from the new dividend:
-6x + 3
-(-6x - 6)
9
Since the degree of the new dividend (the remainder) is lower than the degree of the divisor, we have completed the division.
Therefore, P(x) = -x^3 - 5x + 3 = (x + 1)(-x^2 - 6) + 9.
b) To find the quotient and remainder when dividing P(x) = 27x^5 − 9x^4 + 3x^2 − 3 by D(x) = 3x^2 − 3x + 1, we will use long division.
Long Division:
Step 1: Write the dividend (P(x)) and divisor (D(x)) in descending powers of x:
27x^5 - 9x^4 + 3x^2 - 3 ÷ 3x^2 - 3x + 1
Step 2: Divide the first term of the dividend (27x^5) by the first term of the divisor (3x^2) to get 9x^3 as the first term of the quotient. Multiply this term by the entire divisor and subtract it from the dividend:
27x^5 - 9x^4 + 3x^2 - 3
-(27x^5 - 27x^3 + 9x^2)
-6x^2 - 3
Step 3: Bring down the next term (3x^2) and continue the division:
-6x^2 - 3 ÷ 3x^2 - 3x + 1
Step 4: Divide the first term of the new divisor (-6x^2) by the first term of the divisor (3x^2) to get -2 as the next term of the quotient. Multiply this term by the entire divisor and subtract it from the new dividend:
-6x^2 - 3
-(-6x^2 + 6x - 2)
-9x + 1
Since the degree of the new dividend (the remainder) is lower than the degree of the divisor, we have completed the division.
Therefore, P(x) = 27x^5 − 9x^4 + 3x^2 − 3 = (3x^2 - 3x + 1)(9x^3 - 2x^2 - 9x + 1) - 9x + 1.