(3x^4-5x^3+7)/(x+1)
To simplify the given expression (3x^4-5x^3+7)/(x+1), we can use the polynomial long division method. Here's how you can do it step by step:
Step 1: Arrange the terms of the numerator in descending order of powers of x. So, we have 3x^4 - 5x^3 + 0x^2 + 0x + 7 as the numerator and x + 1 as the denominator.
Step 2: Divide the first term of the numerator (3x^4) by the first term of the denominator (x) to get 3x^3. Write this as the quotient on the left side.
Step 3: Multiply the entire denominator (x + 1) by the quotient obtained in step 2 (3x^3). Now subtract the result from the numerator of the previous step.
(3x^4 - 5x^3 + 0x^2 + 0x + 7) - (3x^3(x + 1))
Step 4: Simplify the result obtained in step 3. Distribute the 3x^3 to both terms in the parentheses and combine like terms.
(3x^4 - 5x^3 + 0x^2 + 0x + 7) - (3x^4 + 3x^3)
= 3x^4 - 5x^3 + 0x^2 + 0x + 7 - 3x^4 - 3x^3
= -8x^3 + 0x^2 + 0x + 7
Step 5: Repeat steps 2-4 with the simplified result (-8x^3 + 0x^2 + 0x + 7) as the new numerator.
Step 2: (-8x^3) / (x) = -8x^2
Step 3: (-8x^3 + 0x^2 + 0x + 7) - (-8x^2(x + 1)) = -8x^3 + 0x^2 + 0x + 7 + 8x^3 + 8x^2 = 8x^2 + 0x + 7
Step 5: Repeat steps 2-4 with the new numerator (8x^2 + 0x + 7).
Step 2: (8x^2) / (x) = 8x
Step 3: (8x^2 + 0x + 7) - (8x(x + 1)) = 8x^2 + 0x + 7 - 8x^2 - 8x = -8x + 7
Finally, we have the simplified expression -8x + 7 as the quotient. Hence, (3x^4-5x^3+7)/(x+1) = 3x^3 - 8x^2 + 8x - 8 + 7/(x+1).