Search results

For (x^2 - 9) / x > 0, either both x^2 - 9 and x are negative, or both are positive, i.e. either x^2 - 9 < 0 and x < 0 or x^2 - 9 > 0 and x > 0. First case yields -3 < x < 0, second case x > 3. Solution set is {x: -3 < x < 0} U {x: x > 3}. :) :) :wave:

Question 2: From my previous post, for n = n, fraction (non-empty) = n/(2n - 1) .: fraction (empty) = 1 - n/(2n - 1). .: number of empty pigeonholes after the distribution of n letters =n[1 - n/(2n - 1)]. .: number of letters in the next round of distribution =n[1 - n/(2n - 1)]...

This is an investigation: For part 1. When n = 1, fraction = 1/1 n = 2, fraction = 2/3 n = 3, fraction = 3/5 n = 4, fraction = 4/7 n = 5, fraction = 5/9 . . n = n, fraction = n/(2n - 1)...

Note that x^2 -3x + 2 =(x - 1)(x - 2) Use the factor theorem to show (x - 1) and (x - 2) are factors of P(x). .: x^2 -3x + 2 is a factor of P(x). :) :) :wave:

An excellent approach based on simple geometry. More suited to clever year 7 students. One has to think outside the triangle! :) :) :wave:

May be you try this one: P(x) = 4x^8 - x^6 + 4x^4 - 4x^3 + 3x^2 + x - 1 is divided by D(x) = x^4 + 1. Find Q(x) and R(x). Find the highest monic common factor of P(x), Q(x) and R(x). :) :) :wave:

Proof by contradiction: Let p+q sqrt(r) be a root of P(x), i.e. [p+q sqrt(r)]^3 - [p+q sqrt(r)] + 3 = 0, expand and collect rational and irrational parts to obtain {p(p^2 +3rq^2 - 1) + 3} + q(3p^2 + rq^2 - 1)sqrt(r) = 0. .: p(p^2 +3rq^2 - 1) + 3 = 0 ........(1) and 3p^2 + rq^2 - 1 = 0...

Actually, it is even easier than you think. It can be considered as motion with constant acceleration. Acceleration down the inclined plane a = gsin10, u = 0, s = 1, v^2 = u^2 + 2as = 2gsin10, v = sqrt(2gsin10) :) :) :wave:

But if you divide e.g. x^3 + ..... by (ax + b), you 'll get (1/a)x^2 + .... :) :) :wave:

e.g. (2x - 1) = 2(x - 1/2) (2x - 1) is non-monic, (x - 1/2) is monic. (3x^2 + 6x - 5) = 3(x^2 + 2x - 5/3) (3x^2 + 6x - 5) is non-monic, (x^2 + 2x - 5/3) is monic. Example P(x) = D(x)Q(x) + R(x) P(x) = (x^4 + x^2 + 3)(4x^4 - 1) + (2x^3 + 2x^2 - x - 1) =(x^4 + x^2 + 3)(2x^2 + 1)(2x^2 - 1) +...

P(x) divided by D(X) results in the quotient Q(x) and remainder R(x) i.e. P(x)/D(x) = Q(x) + R(x)/D(x) .: P(x) = D(x)Q(x) + R(x) .: P(x)/Q(x) = D(x) + R(x)/Q(x) i.e. P(x) divided by Q(X) results in the quotient D(x) and remainder R(x). :) :) :wave:

Try to make a simple one like: http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/indmot.html#c1 Warning! Do not connect directly to the mains supply. Use school lab power pack AC. Make sure electromagnet has many windings in order not to short circuit the power pack. :) :) :wave:

Meaning the angle between the line y =2x + 3 and the tangent to the curve y = x^2 at an intersecting point. Step 1: Find the x-coordinates of the intersecting points. Step 2: Use calculus to find the gradient of the tangent at each intersecting point. Step 3: At each intersecting point use...

b = 2(1 - 2sin10)sin50 = 1 found with a calculator. Without calculator: Use the sine rule to find a = sin20 / sin80 Use the sine rule to find b/sin100 = (1-a)/sin40 b = sin100(1 - sin20 / sin80) /sin40 Since sin100 = sin80, .: b = (sin80 -...

P(x) = (x - a)U(x) Q(x) = (x - a)V(x) P(x) - Q(x) = (x - a)U(x) - (x - a)V(x) = (x - a)[U(x) - V(x)] .: x - a is a factor of P(x) - Q(x). :) :) :wave:

Extend CD until it meets AB at E It can be shown that ACE and BDE are isosceles triangles. Let AB = AC = 1 (without loss of generality), and let AE = a and BD = b. It can be shown that a = 2 sin10deg. .: BE = ED = 1 - a = 1 - 2sin10 .: b = 2(1 - 2sin10)sin50 = 1. ...

This is pretty much like changing an improper fraction to a mixed number: e.g. 5/3 = 1 + 2/3 Your example: (x³ - x² + x + 1)/ (2x - 3) = x²/2 + x/4 + 7/8 + (29/8)/(2x - 3) The mixed form on the RHS is a useful form in some situations, e.g. in finding integral integral[(x³ - x² + x +...

The weight of the rocket (+payload) has to be accounted for: 1.43 x 10^7 - 4.8 x 10^5 X 9.8 = 4.8 x 10^5 a a = 2.0 x 10^1 ms^(-2) :) :) :wave:

It is the x-axis because the parabola is a quadratic function of x. If the quadratic function were not given, there would be two possibilities, one from the x-axis: 8y=x^2+6x+25, and the other from the y-axis: -6x=y^2-8y+25 :) :) :wave:

The edited version is much simpler than the original one that you posted. Couldn't you do such a 'simple' trig equation? :) :) :wave: