Show:
Direct Variation & Proportion
Inverse & Joint Variation
Dividing Polynomials & Synthetic Division
The Remainder & Factor Theorems
Other Useful Theorems
100
y=mx
What is the equation of a direct variation?
100
xy = k
What is the equation of an inverse variation?
100
x3-5x2+4x-2 / (x-2) = x2-3x-2+(-6) / (x-2) Identify the Quotient and the Remainder
What is x2-3x-2 and -6/x-2 ?
100
Consider the following: P(x) = x3-2x2-5x-7 and c=4. By the remainder theorem, P(c) and in this case P(4), is equal to the________ when the above P(x) is divided by x-4.
What is the remainder?
100
1.Every polynomial equation with positive degree n has exactly ______ roots. 2.Conjugate Root Theorem: If a polynomial equation with real coefficients has a+bi as a root, then_______is also a root.
What is n and what is a-bi ?
200
y1/ x1 = y2/ x2
What is the equation for a proportion?
200
If y is inversely proportional to x, and y = 3 when x = 4, find x when y = 6.
What is 2?
200
Divide: x2+3x-4/x+2
What is x+1+(-6)/(x+2)
200
Consider the following: P(x) = x3-2x2-5x-7 and c=4. Using synthetic division, find P(c).
What is 5?
200
Consider the polynomial equation: x4-2x3+4x2+2x-5=0 Determine the number of roots for this equation. Also, the above equation has the following roots: 1, -1, and 1-2i. Based on the Conjugate Root Theorem, identify the remaining root.
What is 4 and what is 1+2i ?
300
If y varies directly as x, and y = 4 when x = 2, find y when x = 1.
What is 2?
300
Name the variation where a quantity varies directly as the product of two or more other quantities. Ex) p = kqr2 where p varies jointly as q and the square of r
What is a joint variation?
300
Synthetic division is an efficient way to divide a polynomial by a binomial of the form x-c. Consider the division of 2x3-4x2-7x+5 by x-3. Identify the coefficients of dividend and c.
What is 2, -4, -7, 5 and what is 3?
300
Consider the following: P(x)= x7-x5+x3-x and x+1. By the factor theorem, the above binomial x+1 is a factor of the above polynomial if P(-1)=0. Is x+1 a factor of the above given polynomial?
What is yes?
300
Consider the following: P(x)=x5+x4-3x2+4x+6. Based on Descartes' Rule of Signs, the number of positive real roots of P(x)=0 is ____or___and the number of negative real roots of P(x)=0 is ____ or____.
What is 2 or 0 and what is 3 or 1?
400
If y is directly proportional to x, and y = 6 when x = 3, find x when y = 2.
What is 1?
400
Write the equation for y varies jointly as r and the square of s and inversely as t.
What is y = krs2/t
400
Divide using synthetic division: x3+3x2-2x-6/(x+3)
What is x2-2 ?
400
Find a polynomial equation with integral coefficients that has the given numbers as roots: 1,2,-3
What is x3-7x+6=0 ?
400
Find a cubic equation with integral coefficients that has 2 and 3-i as roots.
What is x3-8x2+22x-20=0 ?
500
A load of 10 kg stretches a certain spring 4 in. What load would stretch the spring 6 in.?
What is 15 kg?
500
A radio signal frequency varies inversely as the wave length. A signal frequency of 1000 kHz has wavelength 200 m. What frequency has a signal of wavelength 400 m?
What is 500 kHz?
500
Divide using synthetic division: 2z3-z2+8z-4/(z+2i)
What is 2z2-(1+4i)z+2i with remainder 0
500
Solve x3+x+10=0 given that -2 is a root.
What is -2, 1+2i, 1-2i ?
500
Solve x3+x-10=0 given that -1+2i is a root.
What is -1+2i, -1-2i, and 2 ?