Find the quotient and remainder in the following:
a) x^4 +3x^3 + 5x^2 - 6x - 8 is divided by x + 2
x^3 + x^2 + 3x - 12
remainder 16
Thanks mate.
To find the quotient and remainder when dividing a polynomial by another polynomial, we can use the polynomial long division method.
Step 1: Write the given polynomial dividend and the divisor in descending order of powers of x.
Dividend: x^4 + 3x^3 + 5x^2 - 6x - 8
Divisor: x + 2
Step 2: Divide the first term of the dividend (x^4) by the first term of the divisor (x). The result is the first term of the quotient (x^3).
x^3
---------------
x + 2 | x^4 + 3x^3 + 5x^2 - 6x - 8
Step 3: Multiply the divisor (x + 2) by the term of the quotient (x^3), and write the resulting polynomial below the dividend, aligning like terms.
x^3
---------------
x + 2 | x^4 + 3x^3 + 5x^2 - 6x - 8
- (x^4 + 2x^3)
---------------
x^3 + 5x^2 - 6x
Step 4: Subtract the polynomial obtained in step 3 from the dividend to get a new dividend.
x^3
---------------
x + 2 | x^4 + 3x^3 + 5x^2 - 6x - 8
- (x^4 + 2x^3)
---------------
x^3 + 5x^2 - 6x
- (x^3 + 2x^2)
---------------
3x^2 - 6x
Step 5: Repeat steps 2-4 with the new dividend.
Dividend: 3x^2 - 6x
Divisor: x + 2
x^3 + 3
x^3 + 5x^2 - 6x
- (x^3 + 2x^2)
---------------
3x^2 - 6x
- (3x^2 + 6x)
-------------------------------------
0
Step 6: The process stops when the degree of the new dividend is smaller than the degree of the divisor. The last dividend obtained is the remainder.
In this case, the remainder is 0, indicating that the divisor x + 2 fully divides the dividend x^4 + 3x^3 + 5x^2 - 6x - 8.
The quotient is x^3 + 3, and the remainder is 0.