7) $8x^{4}-8x^{3}-x + 1div4x^{2}+2x + 1$
7) $8x^{4}-8x^{3}-x + 1div4x^{2}+2x + 1$
Answer
Explanation:
Step1: Set up long - division
We perform polynomial long - division of (8x^{4}-8x^{3}-x + 1) by (4x^{2}+2x + 1).
Step2: Divide the leading terms
Divide the leading term of the dividend (8x^{4}) by the leading term of the divisor (4x^{2}). (\frac{8x^{4}}{4x^{2}} = 2x^{2}).
Step3: Multiply and subtract
Multiply (4x^{2}+2x + 1) by (2x^{2}): (2x^{2}(4x^{2}+2x + 1)=8x^{4}+4x^{3}+2x^{2}). Subtract this from the dividend: ((8x^{4}-8x^{3}-x + 1)-(8x^{4}+4x^{3}+2x^{2})=-12x^{3}-2x^{2}-x + 1).
Step4: Repeat the process
Divide the leading term of (-12x^{3}-2x^{2}-x + 1) (which is (-12x^{3})) by the leading term of (4x^{2}+2x + 1) (which is (4x^{2})). (\frac{-12x^{3}}{4x^{2}}=-3x). Multiply (4x^{2}+2x + 1) by (-3x): (-3x(4x^{2}+2x + 1)=-12x^{3}-6x^{2}-3x). Subtract this from (-12x^{3}-2x^{2}-x + 1): ((-12x^{3}-2x^{2}-x + 1)-(-12x^{3}-6x^{2}-3x)=4x^{2}+2x + 1).
Step5: Divide the remaining terms
Divide the leading term of (4x^{2}+2x + 1) (which is (4x^{2})) by the leading term of (4x^{2}+2x + 1) (which is (4x^{2})). (\frac{4x^{2}}{4x^{2}} = 1). Multiply (4x^{2}+2x + 1) by (1): (4x^{2}+2x + 1). Subtract this from (4x^{2}+2x + 1), the remainder is (0).
Answer:
(2x^{2}-3x + 1)