Find the quotient and the remainder in the following.
a). x^4 + 3x^3 + 5x^2 - 6x - 8 is divided by x + 2
b). 2x^4 - 3x^3 + 6x^2 - 5x - 4 is divided by 2x - 3
Please help, this is urgent. Thanks.
this is b answer.
x^3 + 3x + 2
remainder 2
Thanks kuai.
To find the quotient and remainder when a polynomial is divided by another polynomial, we can use polynomial long division. Polynomial long division is similar to the long division method used for numbers.
a) To divide x^4 + 3x^3 + 5x^2 - 6x - 8 by x + 2:
Step 1: Write the dividend (x^4 + 3x^3 + 5x^2 - 6x - 8) and the divisor (x + 2) in descending order of the powers of x.
x^4 + 3x^3 + 5x^2 - 6x - 8
÷ (x + 2)
Step 2: Divide the term with the highest power of x from the dividend by the term with the highest power of x from the divisor. Place the result above the line.
x^3
Step 3: Multiply the divisor by the result obtained in Step 2 and write it below the dividend, aligning like terms.
x^3 + 2x^2
___________________
x + 2 ) x^4 + 3x^3 + 5x^2 - 6x - 8
Step 4: Subtract the result obtained in Step 3 from the corresponding terms of the dividend.
x^3 + 2x^2
___________________
x + 2 ) x^4 + 3x^3 + 5x^2 - 6x - 8
- (x^4 + 2x^3) (subtracting term by term)
___________________
x^3 + 5x^2 - 6x - 8
Step 5: Bring down the next term from the dividend (in this case, -6x).
x^3 + 2x^2
___________________
x + 2 ) x^4 + 3x^3 + 5x^2 - 6x - 8
- (x^4 + 2x^3) (subtracting term by term)
___________________
x^3 + 5x^2 - 6x - 8
- (x^3 + 2x^2) (subtracting term by term)
___________________
3x^2 - 6x - 8
Step 6: Repeat steps 2-5 with the new polynomial obtained after subtraction.
x^3 + 2x^2 + 3x
___________________
x + 2 ) x^4 + 3x^3 + 5x^2 - 6x - 8
- (x^4 + 2x^3) (subtracting term by term)
___________________
x^3 + 5x^2 - 6x - 8
- (x^3 + 2x^2) (subtracting term by term)
___________________
3x^2 - 6x - 8
- (3x^2 + 6x) (subtracting term by term)
___________________
-14x - 8
Step 7: Since the degree of the resulting polynomial (-14x - 8) is less than the degree of the divisor (x + 2), we have obtained the remainder.
Therefore, the quotient is x^3 + 2x^2 + 3x, and the remainder is -14x - 8.
b) To divide 2x^4 - 3x^3 + 6x^2 - 5x - 4 by 2x - 3:
The steps for polynomial long division are similar to those in part (a), so you can follow the same procedure to find the quotient and remainder.
I hope this helps! Let me know if you have any further questions.