(3x^4-2x^3+4x-5) divided by (x^2+4)
quotient and remainder
To divide (3x^4 - 2x^3 + 4x - 5) by (x^2 + 4) and find the quotient and remainder, we can use polynomial long division. Here's how to do it step by step:
Step 1: Make sure the divisor (x^2 + 4) is written in descending order. It's already in the correct order in this case.
Step 2: Divide the leading term of the dividend (3x^4) by the leading term of the divisor (x^2). The result is 3x^2.
Step 3: Multiply the quotient obtained in step 2 (3x^2) by the entire divisor (x^2 + 4). The result is 3x^4 + 12x^2.
Step 4: Subtract the result obtained in step 3 from the dividend (3x^4 - 2x^3 + 4x - 5). The subtraction should be done term by term:
(3x^4 - 3x^4) - (2x^3) = -2x^3
(-2x^3) - (-12x^2) = -2x^3 + 12x^2
(12x^2) - (4x) = 12x^2 - 4x
(-4x) - (-5) = -4x + 5
Step 5: Repeat steps 2 to 4 with the new polynomial obtained in step 4 (-2x^3 + 12x^2 - 4x + 5) as the dividend.
Step 6: Divide the leading term of the new dividend (-2x^3) by the leading term of the divisor (x^2). The result is -2x.
Step 7: Multiply the quotient obtained in step 6 (-2x) by the entire divisor (x^2 + 4). The result is -2x^3 - 8x.
Step 8: Subtract the result obtained in step 7 from the new dividend (-2x^3 + 12x^2 - 4x + 5):
(-2x^3 - (-2x^3)) - (12x^2) = -12x^2
(12x^2) - (-8x) = 12x^2 + 8x
(8x) - (4x) = 4x
(4x) - 5 = 4x - 5
Step 9: Repeat steps 2 to 4 with the new polynomial obtained in step 8 (-12x^2 + 8x + 4x - 5) as the dividend.
Step 10: Divide the leading term of the new dividend (-12x^2) by the leading term of the divisor (x^2). The result is -12.
Step 11: Multiply the quotient obtained in step 10 (-12) by the entire divisor (x^2 + 4). The result is -12x^2 - 48.
Step 12: Subtract the result obtained in step 11 from the new dividend (-12x^2 + 8x + 4x - 5):
(-12x^2 - (-12x^2)) - 8x = -8x
(8x) - (-48) = 8x + 48
(48) - 5 = 43
At this point, we have reached the end of the polynomial long division process. The quotient is 3x^2 - 2x - 12, and the remainder is 8x + 43.