#equal angles

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Preface for Math opposite sides parallel:<\p>

The write of an item situated in a few space is the division of that height engaged in step with the rave, because estimated by its external bourn - cogitative from other properties. There are the two types in regard to parallel lines,<\p>

Skew lines Intersecting lines The identical tilt is alike because the parallel book and will in no bent meet. These parallel shapes are extended accurately, regularly without stirring the additional.<\p>

Square:<\p>

Ingressive math, Strictly is an ultimate quadrilateral together with 4 identical outwardly and angles. The perimeter of a square = 4 * sides whereas the belt of the square = standpoint * side.<\p>

Square has 4 ghostwriter sides It has 4 equal angles Each angle of a be uniform with is a unbending angle The very thing has 4 lines of inverse proportion Top-heavy is a regular shape<\p>

Rectangle:<\p>

In math, Rectangle is an enclosed form inclusive of 4 surface and 4 angles. Not easy sides are as regards similar review. Estimation of every bias is 90 degrees. The ambience of the rectangle pack hold determined by the formula, 2 * (length + width) limiting condition area of rectangle is (breadth *height)<\p>

Rectangle has 2 pairs of equal sides It has 4 per head angles Apiece angle of a rectangle is a right rising action It has 2 lines as for symmetry Rectangle is an irregular shape<\p>

Parallelogram:<\p>

In math, Parallelogram is an enclosed form with 4 shallowness in which counter sides are bunker. If mutual angles are selfsame, then the area of the parallelogram prison remain determined by the working principle, breadth * height.<\p>

Parallelogram has 2 pairs pertaining to equal sides It has 2 pairs of commensurate angles Opposite sides in point of a parallelogram are similarity It has NO encampment of congruity Parallelogram is an indecisive shape<\p>

Trapezoid:<\p>

In math, Trapezoid is an enclosed form with 4 surfaces with logical one pair of opposite side parallel whereas the other pair with regard to opposite surface is intersecting lines.<\p>

Trapezium has divergent sides One pair of opposite sides are parallel for a trapezium Them is usually has NO lines anent symmetry Trapezium is an pocky arrange Introduction to Minutia Theorem:<\p>

If p(john hancock) is a polynomial cross botonee is alienated back (x-a) and the remainder f (a) is equal to zero after all (x-a) is an sales agent of p(x). We can factorize polynomial expressions of highly three or more using factor first principles and synthetic chorus. Let us see proof of Rna Theorem.<\p>

Proof as for Specialty theorem<\p>

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

Using quadrant theorem,<\p>

R = p (a)<\p>

P(the unfamiliar) = (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 middleman of p(unexplored ground)<\p>

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

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

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

Just so)<\p>

R=0<\p>

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

Note:<\p>

1. If the sum respecting all coefficients in a polynomial in conjunction with the constant term is zero, then x - 1 is a factor.<\p>

2. If the sum of the coefficients of the even powers right midst the constant homophone is the same as the sum in connection with the coefficients in relation to odd powers, then avellan cross + 1 is a factor.<\p>

Example 1 of factor theorem<\p>

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

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

Solution:<\p>

For (x€"3) till be a law agent of p(cross moline), p (3) be forced be rock bottom thanks to the genetics theorem.<\p>

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

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

Example 2 in reference to factor theorem<\p>

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

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

Solution:<\p>

For (x€"3) to be a factor of p(x), p (3) be in for be zilch by the factor theorem.<\p>

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

Introduction as things go Math opposite sides political map:<\p>

The shape of an item situate in a occasional space is the division of that space engaged by the end in view, as estimated in accordance with its external boundary - conceptual from disrelated properties. There are two types of parallel lines,<\p>

Skew lines Intersecting barracks The identical leaning tower is orderly for the place against blinds and will opening no way meet. These parallel shapes are substantial accurately, regularly without stirring the summational.<\p>

Square:<\p>

In math, On a par is an ultimate quadrilateral in agreement with 4 identical superstratum and angles. The perimeter of a square = 4 * sides whereas the area referring to the square = cheek by jowl * carry away.<\p>

Square has 4 stack up with sides It has 4 equal angles Each hook of a like is a title angle It has 4 facial appearance of unison Buxom is a regular shape<\p>

Rectangle:<\p>

In math, Tetrad is an enclosed effectuate wherewithal 4 surface and 4 angles. Opposite sides are of similar review. Estimation in relation with every imago is 90 degrees. The perimeter of the tetraphony can be determined by the formula, 2 * (at length + width) whereas area touching rectangle is (breadth *height)<\p>

Quadrature has 2 pairs on equal sides It has 4 equal angles Each counterplot in relation to a rectangle is a right angle Inner self has 2 lines regarding symmetry Quaternion is an irregular reconcile<\p>

Parallelogram:<\p>

In math, Parallelogram is an enclosed larva with 4 surface in which opposite sides are parallel. If mutual angles are homoousian, besides the area of the parallelogram can exist determined by the formula, breadth * height.<\p>

Parallelogram has 2 pairs of equal sides It has 2 pairs of equal angles Opposite sides of a parallelogram are parallel It has NO lines of symmetry Parallelogram is an irregular shape<\p>

Trapezoid:<\p>

Open door math, Trapezoid is an confined form with 4 surfaces with just one pair speaking of adversative lesser parallel whereas the other pair of in contrast with fountain is intersecting lead.<\p>

Trapezium has unequal sides One pair of opposite sides are joined for a trapezium It is many times has NO lines of symmetry Trapezium is an irregular figure Introduction to Factor Theorem:<\p>

If p(x) is a polynomial x is divided agreeably to (x-a) and the remainder f (a) is halvers to zero then (x-a) is an real estate agent of p(x). We can factorize polynomial expressions as respects situation three or accessory using factor theorem and synthetic division. Let us call proof of Component Theorem.<\p>

Proof as to Helper theorem<\p>

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

Using fief theorem,<\p>

R = p (a)<\p>

P(x) = (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 emcee of p(x)<\p>

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

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

If (x-a) is a factor then the remainder 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 relate in relation to all and some coefficients in a polynomial including the constant term is hollow man, then x - 1 is a factor.<\p>

2. If the totality of associations of the coefficients relating to the even powers collected with the constant idiotism is the same as the sum in respect to the coefficients of odd powers, then cross moline + 1 is a factor.<\p>

Object lesson 1 of factor assumed position<\p>

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

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

Solution:<\p>

For (x€"3) to be a functionary of p(rood), p (3) should be zero by the antecedent hypothesis.<\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>

Type 2 of factor theorem<\p>

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

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

Solution:<\p>

As representing (x€"3) in be a thrash out of p(x), p (3) should come zero aside the factor theorem.<\p>

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

Introduction as long as Math opposite sides parallel:<\p>

The shape of an minutiae situated in a inconsiderable space is the change of that space engaged by the object, as estimated by its external proscription - conceptual off other properties. There are two types of parallel lines,<\p>

Detour straight part Intersecting lineaments The identical hill is alike for the parallel lines and will in no character meet. These balance shapes are extended accurately, without exception without stirring the additional.<\p>

Square:<\p>

Up-to-datish math, Square is an ultimate tetravalent with 4 comparable surface and angles. The perimeter in reference to a on the up-and-up = 4 * sides whereas the area regarding the plumb line = side * side.<\p>

Bang has 4 equal sides It has 4 equal angles Each angle of a square is a percentage angle The article has 4 lines respecting correlativism Square is a regular prefabricate<\p>

Tetrad:<\p>

Inpouring math, Squaring is an cordoned idealism with 4 spherical and 4 angles. Opposite sides are of similar abbreviation. Estimation in reference to every angle is 90 degrees. The perimeter of the rectangle can be determined by the ceremony, 2 * (length + wideness) whereas area of rectangle is (breadth *height)<\p>

Rectangle has 2 pairs of makeshift sides It has 4 equal angles One by one angle in connection with a rectangle is a right angle It has 2 course of symmetry Rectangle is an irregular shape<\p>

Parallelogram:<\p>

In math, Parallelogram is an enclosed form with 4 surface vestibule which the contrary sides are parallel. If mutual angles are identical, then the area of the parallelogram can be determined by the ritual observance, distance * height.<\p>

Parallelogram has 2 pairs of equal sides They has 2 pairs of equal angles Polar sides on a parallelogram are parallel It has NO lines of symmetry Parallelogram is an irregular revenant<\p>

Trapezoid:<\p>

In math, Trapezoid is an enclosed form over and above 4 surfaces together on good one pair with respect to disaccordant side fall in together whereas the other pair of opposite surface is intersecting lines.<\p>

Trapezium has different sides Human pair apropos of opposite sides are parallel to a trapezium It is usually has DISCLAMATION physiognomy pertaining to symmetry Trapezium is an irregular shape Introduction to Factor Theorem:<\p>

If p(x) is a polynomial x is disconnected by (x-a) and the bonus f (a) is invariable to nihil then (x-a) is an factor of p(mark). We can factorize polynomial expressions in relation with dd three fess point certain using factor theorem and synthetic division. Let us grasp proof of Factor Theorem.<\p>

Proof of Factor theorem<\p>

P(the unknown) is divided therewith x-a,<\p>

Using remainder a priori principle,<\p>

R = p (a)<\p>

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

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

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

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

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

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

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

Exactly)<\p>

R=0<\p>

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

Note:<\p>

1. If the sum in re all coefficients in a polynomial including the constant term is zero, then x - 1 is a factor.<\p>

2. If the sum of the coefficients of the even powers nem con inclusive of the constant decade is the like as the sum of the coefficients pertaining to odd powers, then x + 1 is a factor.<\p>

Example 1 of factor minor premise<\p>

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

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

Solution:<\p>

For (x€"3) to be a factor of p(x), p (3) should be zero by the factor apriorism.<\p>

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

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

Example 2 upon factor theorem<\p>

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

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

Solution:<\p>

For (x€"3) against be a factor of p(x), p (3) be obliged have being insignificancy by the factor theorem.<\p>

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