Answer :
Polynomial long division is used to divide[tex]x^4[/tex] - [tex]x^3[/tex] - [tex]70x^2[/tex]+ x + 6 by [tex](x - 1)^2[/tex]. The step-by-step process involves dividing terms by the highest power from the divisor and subtracting the obtained products from the polynomial. Finally, we are left with a quotient and a remainder. Therefore, the final division is: (x4 - x3 - 70x2 + x + 6) ÷ (x - 1)2 = x3 - 70x + 6 - (69x - 6) / (x - 1)2
To divide the polynomial x4 - x3 - 70x2 + x + 6 by the expression (x - 1)2, we use polynomial long division. First, arrange the polynomial in descending order of powers. Then, divide the terms by the highest power from the divisor, one at a time. Finally, perform the long division process until the remainder is a lower degree than the divisor.
Using polynomial long division, we divide by the factor (x - 1) twice. Here is the step-by-step process:
Write the polynomial in descending order of powers: x4 - x3 - 70x2 + x + 6
Divide the first term of the polynomial by the first term of the divisor: x4 ÷ x = x3
Multiply the entire divisor by the obtained quotient: (x - 1) * x3 = x4 - x3
Subtract the obtained product from the polynomial: (x4 - x3 - 70x2 + x + 6) - (x4 - x3) = -70x2 + x + 6
Repeat steps 2-4 until the remainder is a lower degree than the divisor. In this case, we repeat the process one more time:
Divide the first term of the current polynomial by the first term of the divisor: -70x2 ÷ x = -70x
Multiply the entire divisor by the obtained quotient: [tex](x - 1) * (-70x)[/tex]= -70x2 + 70x
Subtract the obtained product from the current polynomial: (-70x2 + x + 6) - (-70x2 + 70x) = -69x + 6
The remainder is -69x + 6, which is a lower degree than the divisor (x - 1)2. Therefore, the final division is: (x4 - x3 - 70x2 + x + 6) ÷ (x - 1)2 = x3 - 70x + 6 - (69x - 6) / (x - 1)2
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