Questions: What are the quotient and remainder when 3x^4-x^2 is divided by x^3-x^2+2?

Transcript text: What are the quotient and remainder when $3 x^{4}-x^{2}$ is divided by $x^{3}-x^{2}+2 ?$

Solution

Solution Steps

Step 1: Identify the leading terms

The leading term of the dividend is $3x^4$ and of the divisor is $x^3$.

Step 2: Division process

Quotient term 1: The leading term of the dividend divided by the leading term of the divisor gives us a quotient term of $0.25x^1$.

Quotient term 2: The leading term of the dividend divided by the leading term of the divisor gives us a quotient term of $-0.5x^0$.

Step 3: Subtract and repeat

Subtract the product of the divisor and the current quotient term from the dividend, and repeat the process with the remainder as the new dividend until the degree of the remainder is less than the degree of the divisor.

Final Answer:

Quotient polynomial: $Q(x) = 0.25x^1 - 0.5x^0$ Remainder polynomial: $R(x) = 2.75x^4 + 0.75x^3 - 1.5x^2 - 0.5x^1 + 1.0x^0$

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