StudentAidUnit 9: Division of Polynomials — Exercise 9 2
11th Class Mathematics · Unit 9: Division of Polynomials
1.Consider a data set at monthly sales figures. A polynomial regression model P(x) = x³ + 2x² + x - 3 is fitted to this data. If the observed sales in the 5th month are 240 units. Find the percentage error.
Given
Observed sales
Observed = 240 units
Formula
Error formula
Error = Observed - Predicted
Given
Polynomial model
P(x) = x3 + 2x2 + x - 3
Working
Predict sales for 5th month
P(5) = (5)3 + 2(5)2 + 5 - 3
P(5) = 125 + 50 + 5 - 3
P(5) = 177
Calculate error
Error = 240 - 177 = 63
Formula
Percentage error
63240 × 100
Result
Percentage error
= 26.25%
2.A retailer company has developed a polynomial regression model to predict weekly product demand: D(w) = w³ - 2w² + 5w - 4, where D(w) represents predicted demand (in units) and w is the week number. Use the remainder theorem to predict demand for 3rd week. If the observed demand is 22 units, calculate the prediction error.
Given
Polynomial model
D(w) = w3 - 2w2 + 5w - 4
Week number
w = 3
Observed demand
Observed demand = 22 units
Formula
Remainder theorem
D(3) = (3)3 - 2(3)2 + 5(3) - 4
D(3) = 27 - 18 + 15 - 4
D(3) = 20
Result
Predicted demand
Predicted demand = 20 units
Working
Prediction error
Error = 22 - 20
Result
Prediction error
= 2 units
3.A digital signal processing system has transfer function with a numerator B(z) = z² - z - 2. Use the factor theorem to find the zeros of the system.
Given
Transfer function numerator
B(z) = z2 - z - 2
Formula
Find zeros
B(z) = 0 ⇒ z2 - z - 2 = 0
Factor
z2 - 2z + z - 2 = 0
z(z - 2) + 1(z - 2) = 0
(z - 2)(z + 1) = 0
Working
Solve
z - 2 = 0; quad z + 1 = 0
Result
Zeros
z = 2, quad z = -1
4.A signal process system has a transfer function H(z) = (z² + 3z + 2)/(z² - 0.2z + 0.9). Find the zero(s) of the transfer function by using factor theorem.
Given
Transfer function
H(z) = z2 + 3z + 2z2 - 0.2z + 0.9
Formula
Find zeros
H(z) = 0 ⇒ z2 + 3z + 2 = 0
Factor numerator
z2 + 2z + z + 2 = 0
z(z + 2) + 1(z + 2) = 0
(z + 2)(z + 1) = 0
Working
Solve
z + 2 = 0; quad z + 1 = 0
Result
Zeros
z = -2, quad z = -1
5.A signal process system has a transfer function H(z) = (z² - 0.5z - 0.5)/(z³ + 1). Find the zero(s) of the transfer function by using factor theorem.
Given
Transfer function
H(z) = z2 - 0.5z - 0.5z3 + 1
Formula
Find zeros
H(z) = 0 ⇒ z2 - 0.5z - 0.5 = 0
Multiply by 2
z2 - 12z - 12 = 0
2z2 - z - 1 = 0
Factor
2z2 - 2z + z - 1 = 0
2z(z - 1) + 1(z - 1) = 0
(2z + 1)(z - 1) = 0
Working
Solve
2z + 1 = 0 or z - 1 = 0
Result
Zeros
z = -0.5, quad z = 1
6.A signal processing system has a transfer function with a denominator A(z) = z² - 0.3z - 0.4. Use factor theorem to find the poles of the system and determine if the system is stable.
Given
Denominator polynomial
A(z) = z2 - 0.3z - 0.4
Formula
Find poles
A(z) = 0 ⇒ z2 - 0.3z - 0.4 = 0
Convert to integer coefficients
z2 - 310z - 410 = 0
10z2 - 3z - 4 = 0
Factor
10z2 + 5z - 8z - 4 = 0
5z(2z + 1) - 4(2z + 1) = 0
(2z + 1)(5z - 4) = 0
Working
Solve
2z + 1 = 0 or 5z - 4 = 0
Result
Poles
z = -0.5, quad z = 0.8
Formula
Stability condition
|z| < 1
Check
|-0.5| = 0.5 < 1 and |0.8| = 0.8 < 1
Result
System stability
Both poles inside unit circle; system is stable
7.The denominator of signal processing system's transfer function equal to A(z) = z² + 1.2z + 0.35. Use factor theorem to determine the location of the corresponding poles and assess the stability of the system.
Given
Denominator polynomial
A(z) = z2 + 1.2z + 0.35
Formula
Find poles
A(z) = 0 ⇒ z2 + 1.2z + 0.35 = 0
Convert to integer coefficients
z2 + 1210z + 35100 = 0
100z2 + 120z + 35 = 0
Factor out 5
5(20z2 + 24z + 7) = 0
20z2 + 24z + 7 = 0
Factor
10z(2z + 1) + 7(2z + 1) = 0
(2z + 1)(10z + 7) = 0
Working
Solve
2z + 1 = 0; quad 10z + 7 = 0
Result
Poles
z = -0.5, quad z = -0.7
Formula
Stability condition
|z| < 1
Check
|-0.5| = 0.5 < 1 and |-0.7| = 0.7 < 1
Result
System stability
Both poles inside unit circle; system is stable