Perform the indicated operation.
$\frac{-12 w^4+22 w^3+28 w^2-53 w+20}{6 w^2-8 w+3}$
To perform the division of the polynomial \(-12 w^4 + 22 w^3 + 28 w^2 - 53 w + 20\) by \(6 w^2 - 8 w + 3\), we use the process of long division for polynomials. Our goal is to express the dividend as the product of the divisor and a quotient plus a remainder, in the form:
\[D(w) = Q(w)d(w) + R(w)\]
where \(D(w)\) is the dividend, \(d(w)\) is the divisor, \(Q(w)\) is the quotient, and \(R(w)\) is the remainder. The degree of the remainder \(R(w)\) must be strictly less than the degree of the divisor \(d(w)\), which is 2.
We divide the leading term of the dividend \(-12w^4\) by the leading term of the divisor \(6w^2\):
\[\frac{-12w^4}{6w^2} = -2w^2.\]
This is the first term of our quotient. Next, we multiply this term by the entire divisor \(6w^2 - 8w + 3\):
\[-2w^2(6w^2 - 8w + 3) = -12w^4 + 16w^3 - 6w^2.\]
Subtracting this result from the dividend gives the new polynomial:
\[(-12w^4 + 22w^3 + 28w^2 - 53w + 20) - (-12w^4 + 16w^3 - 6w^2) = 6w^3 + 34w^2 - 53w + 20.\]
Now, divide the leading term of the new polynomial \(6w^3\) by the leading term of the divisor \(6w^2\):
\[\frac{6w^3}{6w^2} = w.\]
This is the second term of our quotient. Multiplying \(w\) by the divisor \(6w^2 - 8w + 3\) yields:
\[w(6w^2 - 8w + 3) = 6w^3 - 8w^2 + 3w.\]
Subtracting this from our current polynomial result:
\[(6w^3 + 34w^2 - 53w + 20) - (6w^3 - 8w^2 + 3w) = 42w^2 - 56w + 20.\]
Finally, divide the leading term of the result \(42w^2\) by the leading term of the divisor \(6w^2\):
\[\frac{42w^2}{6w^2} = 7.\]
This is the last term of the quotient. Multiplying \(7\) by the divisor gives:
\[7(6w^2 - 8w + 3) = 42w^2 - 56w + 21.\]
Subtracting this from the previous result gives:
\[(42w^2 - 56w + 20) - (42w^2 - 56w + 21) = -1.\]
The remainder is \(-1\). Thus, the quotient is \(-2w^2 + w + 7\) and the remainder is \(-1\). The final expression is:
\[-2w^2 + w + 7 - \frac{1}{6w^2 - 8w + 3}.\]
Polynomial division is essential in control theory and signal processing, where it is used to simplify transfer functions or analyze system stability. For example, if you are designing a filter to process audio signals, the transfer function might be a ratio of two polynomials. Performing long division can help you isolate the steady-state component of the response from the transient or error terms, allowing for more precise control and adjustments in circuit design.