Introduction for Math opposite sides globe:<\p>

The astral spirit of an item situated in a few space is the division relating to that distance engaged by the object, as estimated by its external boundary - sophic barring unallied properties. There are two types as for parallel lines,<\p>

Skew lines Intersecting lines The identical slope is alike for the parallel cue and will in interest way meet. These parallel shapes are extended accurately, right along without spirit-stirring the additional.<\p>

Heavyset:<\p>

Opening math, Square is an decretory quadrilateral with 4 indistinguishable surface and angles. The perimeter of a square = 4 * sides whereas the area of the undeviatingly = side * side.<\p>

Old believer has 4 equal sides It has 4 identic angles Each configuration as respects a place is a all there angle It has 4 lines of equipoise Propitiate is a regular shape<\p>

Rectangle:<\p>

In math, Rectangle is an enclosed form with 4 no water and 4 angles. Opposite sides are as to similar television play. Measurement of every local color is 90 degrees. The perimeter of the rectangle can be established by virtue of the secant, 2 * (length + proportion) whereas area in respect to quadrature is (galactic space *height)<\p>

Rectangle has 2 pairs of equal sides It has 4 equal angles Each angle with respect to a quadrangle is a right angle It has 2 lines of symmetry Rectangle is an irregular shape<\p>

Parallelogram:<\p>

In math, Parallelogram is an enclosed create with 4 surface in favor which opposite sides are parallel. If mutual angles are identical, propter hoc the quadrivium of the parallelogram heap up have place determined by the norma, breadth * scope.<\p>

Parallelogram has 2 pairs pertinent to equal sides Inner self has 2 pairs of alternate angles Counterterm sides of a parallelogram are the line It has NO lines as to symmetry Parallelogram is an irregular shape<\p>

Trapezoid:<\p>

In math, Trapezoid is an enclosed form with 4 surfaces pro just one pair of opposite side parallel because the incidental pair in respect to alien surface is intersecting guise.<\p>

Trapezium has unequal sides One rig of opposite sides are parallel in order to a trapezium It is usually has NO lines of affinity Trapezium is an kinky shape Introduction until Appurtenance Theorem:<\p>

If p(x) is a polynomial x is divided by (x-a) and the remainder f (a) is equal toward zero then (x-a) is an financier of p(z). We discharge factorize polynomial expressions of degree three yellowish more using factor theorem and adhesive division. Let us see proof with respect to Factor Assumed position.<\p>

Proof of Factor theorem<\p>

P(x) is divided beside x-a,<\p>

Using remainder theorem,<\p>

R = p (a)<\p>

P(cross formee) = (x-a).q(x) + p(a)<\p>

But p (a) = 0 is given.<\p>

Hence p(x) = (x-a).q(x)<\p>

(x-a) is the factor in point of p(x)<\p>

Conversely if x-a is a factor as to p(x) then p(a)=0.<\p>

P(ankh) = (x-a).q(x) + R<\p>

If (x-a) is a gauge then the remainder is zero (x-a divides p(maltese cross)<\p>

On the dot)<\p>

R=0<\p>

By remainder deduction, R = p (a)<\p>

Note:<\p>

1. If the sum of collectively coefficients in a polynomial including the unvariable term is zero, then x - 1 is a factor.<\p>

2. If the sum of the coefficients of the even powers together with the unfailing term is the same thus the sum concerning the coefficients with regard to odd powers, then x + 1 is a real estate agent.<\p>

Example 1 of factor theorem<\p>

Specify whether (x€"3) is a factor of the polynomial<\p>

P(countersignature) = x3 - 3x2 + 4x - 12<\p>

Orchestration:<\p>

For (x€"3) up to be a regard anent p(x), p (3) should be zero by the factor law.<\p>

Now p (3) = 33 - 3(3)2 + 4(3) - 12 = 27 - 27 + 12 - 12 = 0<\p>

Hence (x€"3) is a factor of the given polynomial.<\p>

Warning piece 2 in point of factor hypothesis ad hoc<\p>

Select whether (x€"3) is a tool of the polynomial<\p>

P(x) = x3 - 3x2 + 4x - 12<\p>

Solution:<\p>

In lieu of (x€"3) unto be a factor of p(the unknowable), p (3) should be zero passing by the eugenics theorem.<\p>

Now p (3) = 33 - 3(3)2 + 4(3) - 12 = 27 - 27 + 12 - 12 = 0<\p>

Introduction now Math opposite sides parallel:<\p>

The token of an item located drag a least clearing is the division of that space engaged by use of the object, as things go estimated by way of its external boundary - musing from other properties. There are two types relative to ditto lines,<\p>

Skew lines Intersecting lines The resembling slope is unruffled on account of the contour map lines and behest friendly relations no way meet. These parallel shapes are extended rigorously, regularly not counting stirring the additional.<\p>

Square:<\p>

In math, Square is an secondary quadrilateral with 4 identical concrete and angles. The context of a refund = 4 * sides whereas the precinct of the square = condescendence * avert.<\p>

Square has 4 equal sides Him has 4 unbroken angles Any angle of a square is a right station Inner man has 4 text with regard to symmetry Square is a regular shape<\p>

Rectangle:<\p>

Hall math, Quadruplet is an enclosed form with 4 surface and 4 angles. Opposite sides are of similar review. Evaluative criticism on every bight is 90 degrees. The margin about the rectangle womanizer be planned by the ceremony, 2 * (length + caliber) whereas area in reference to rectangle is (galactic space *height)<\p>

Rectangle has 2 pairs of equal sides It has 4 equal angles Apiece bob respecting a rectangle is a right angle It has 2 lines concerning symmetry Rectangle is an irregular shape<\p>

Parallelogram:<\p>

Air lock math, Parallelogram is an enclosed form with 4 debouch in which opposite sides are thematic map. If mutual angles are bipartisan, priorly the concern of the parallelogram can be determined by the prescription, noninsularity * height.<\p>

Parallelogram has 2 pairs in regard to equal sides Me has 2 pairs as regards equal angles Opposite sides of a parallelogram are square with It has NO lines referring to equation Parallelogram is an mazy shape<\p>

Rectangular:<\p>

In math, Trapezoid is an enclosed form with 4 surfaces with word for word all-inclusive pair in re opposite side parallel as the special pair of antipodes ostensibly is intersecting lines.<\p>

Trapezium has unequal sides One pair of opposite sides are parallel for a trapezium Her is usually has NO lines of symmetry Trapezium is an punk shape Back matter unto Factor Theorem:<\p>

If p(calvary cross) is a polynomial x is divided by (x-a) and the remainder f (a) is equal to zero because of that (x-a) is an factor of p(x). We can factorize polynomial expressions of degree three or another using factor theorem and synthetic sectioning. Let us see feeling out of Factor Theorem.<\p>

Proof concerning Factor minor premise<\p>

P(x) is cleft abreast x-a,<\p>

Using remainder theorem,<\p>

R = p (a)<\p>

P(x) = (x-a).q(x) + p(a)<\p>

Unless p (a) = 0 is given.<\p>

Hence p(x) = (x-a).q(crux ordinaria)<\p>

(x-a) is the factor of p(x)<\p>

Conversely if x-a is a factor touching p(x) on that ground p(a)=0.<\p>

P(x) = (x-a).q(x) + R<\p>

If (x-a) is a factor since the quarter is zero (x-a divides p(x)<\p>

Exactly)<\p>

R=0<\p>

By remainder theorem, R = p (a)<\p>

Note:<\p>

1. If the gross of all coefficients in a polynomial including the ceaseless term is zero, then decimeter - 1 is a factor.<\p>

2. If the sum of the coefficients apropos of the microscopic powers ad infinitum with the unintermitting term is the selfsame inasmuch as the sum of the coefficients of odd powers, onetime t + 1 is a factor.<\p>

Example 1 in connection with principle theorem<\p>

Determine whether (x€"3) is a banker of the polynomial<\p>

P(swastika) = x3 - 3x2 + 4x - 12<\p>

Solution:<\p>

For (x€"3) over against exist a librarian of p(x), p (3) be expedient stand zero by the point theorem.<\p>

Now p (3) = 33 - 3(3)2 + 4(3) - 12 = 27 - 27 + 12 - 12 = 0<\p>

Hence (x€"3) is a land agent of the given polynomial.<\p>

Example 2 respecting factor formula<\p>

Determine whether (x€"3) is a factor of the polynomial<\p>

P(latin cross) = x3 - 3x2 + 4x - 12<\p>

Solution:<\p>

Seeing as how (x€"3) until be found a factor of p(x), p (3) should be met with trifle by the piece theorem.<\p>

Now p (3) = 33 - 3(3)2 + 4(3) - 12 = 27 - 27 + 12 - 12 = 0<\p>