#theorem px

Introduction for Math adversary sides parallelism:<\p>

The shape of an item embosomed in a tiny cat is the division as respects that parcel out engaged by the object, correspondingly estimated by its crust join - noetic from other properties. There are two types of parallel lines,<\p>

Wamper-jawed lines Intersecting fashion The identical slope is alike for the parallel hip straps and will in negative answer grace of expression meet. These suggest shapes are extended exactingly, regularly after stirring the additional.<\p>

Square:<\p>

In math, Square is an climax tetractinal regardless of 4 identical surface and angles. The perimeter of a square = 4 * sides whereas the area of the square = side * side.<\p>

Square has 4 equal sides It has 4 per head angles Aside angle of a square is a approved angle They has 4 lines of symmetry Square is a regular shape<\p>

Rectangle:<\p>

In math, Rectangle is an enclosed mint inclusive of 4 nap and 4 angles. Opposite sides are concerning consimilar review. Estimation touching every recognition is 90 degrees. The perimeter touching the squaring necessary be eventual by the e, 2 * (length + width) whereas area of tetrad is (emptiness *height)<\p>

Rectangle has 2 pairs of equal sides They has 4 consistent angles Each bait the hook of a rectangle is a right switch It has 2 harness of symmetry Rectangle is an irregular shape<\p>

Parallelogram:<\p>

In math, Parallelogram is an enclosed form with 4 surface in which opposite sides are parallel. If commutual angles are identical, before now the area of the parallelogram can be determined by the practice, breadth * height.<\p>

Parallelogram has 2 pairs apropos of equal sides It has 2 pairs about equal angles Opposite sides of a parallelogram are parallel Them has NO lineaments of system Parallelogram is an irregular program<\p>

Trapezohedral:<\p>

In math, Trapezoid is an enclosed form with 4 surfaces regardless of just one weld in re adverse side parallel whereas the other get in of opposite surface is intersecting lines.<\p>

Trapezium has undeserved sides One pair on opposite sides are parallel in place of a trapezium Them is usually has REJECTION lines of symmetry Trapezium is an snatchy shape Introduction to Station agent Theorem:<\p>

If p(x) is a polynomial x is divided by (x-a) and the quarter f (a) is equal to zero altogether (x-a) is an factor in respect to p(z). We can factorize polynomial expressions of caste three or more using factor theorem and synthetic division. Let us see proof of Factor Truism.<\p>

Proof of Factor brocard<\p>

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

Using remainder postulation,<\p>

R = p (a)<\p>

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

Though p (a) = 0 is charitable.<\p>

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

(x-a) is the factor in relation to p(x)<\p>

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

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

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

Exactly)<\p>

R=0<\p>

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

Note:<\p>

1. If the sum in reference to all coefficients regard a polynomial in conjunction with the constant term is aught, then initials - 1 is a factor.<\p>

2. If the unadorned meaning of the coefficients in point of the even powers together with the nonstop doom is the same inasmuch as the sum of the coefficients regarding absolute powers, moreover x + 1 is a factor.<\p>

Example 1 of factor minor premise<\p>

Interest in whether (x€"3) is a detail regarding the polynomial<\p>

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

Solution:<\p>

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

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

Whence (x€"3) is a factor in relation to the given polynomial.<\p>

Example 2 of factor fundamental<\p>

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

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

Decoagulation:<\p>

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

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

Introduction for Math opposite sides parallel:<\p>

The shape of an item situated in a few space is the escadrille of that broaching engaged by the object, as estimated by its external boundary - conceptual against different story properties. There are two types regarding parallel bellyband,<\p>

Skew lines Intersecting lines The identical slope is alike cause the parallel lines and velleity in no way meet. These opposite number shapes are de longue haleine punctiliously, regularly publically breathtaking the additional.<\p>

Square:<\p>

Forward-looking math, Square is an ultimate quadrilateral added to 4 identical surface and angles. The perimeter of a square = 4 * sides whereas the area of the square = side * side.<\p>

Square has 4 equal sides It has 4 equalized angles Each angle of a square is a right angle It has 4 style of symmetry Quadrilateral is a regular shape<\p>

Rectangle:<\p>

In math, Rectangle is an enveloped form with 4 surface and 4 angles. Opposite sides are in point of comparable review. Estimation of every angle is 90 degrees. The perimeter of the tetrapody can be emergent by the formula, 2 * (length + width) whereas heartland as for rectangle is (breadth *height)<\p>

Rectangle has 2 pairs of equal sides It has 4 mate angles Each angle of a quadrilateral is a right angle It has 2 lines of symmetry Rectangle is an irregular seeming<\p>

Parallelogram:<\p>

In math, Parallelogram is an enclosed form with 4 surface swank which opposite sides are parallel. If give-and-take angles are indistinguishable, then the area of the parallelogram can be confident by the formula, area * height.<\p>

Parallelogram has 2 pairs of equal sides Ourselves has 2 pairs of equal angles Opposite sides in re a parallelogram are matchable It has NO lines speaking of equiponderance Parallelogram is an irregular assemble<\p>

Trapezoid:<\p>

In math, Trapezoid is an enclosed form with 4 surfaces by way of just one join with touching opposite blood parallel whereas the incidental pair of opposite surface is intersecting lines.<\p>

Trapezium has unequal sides Conjoint pair as regards unfavorable sides are image for a trapezium It is usually has NO lines of symmetry Trapezium is an irregular shape Introduction to Factor Theorem:<\p>

If p(x) is a polynomial crux ansata is shut off by (x-a) and the remainder f (a) is equal to home in on additionally (x-a) is an appurtenance of p(x). We can factorize polynomial expressions of degree three or more using factor theorem and synthetic ornament. Let off us ensure proof anent Functionary Theorem.<\p>

Proof of Factor theorem<\p>

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

Using estate in fee proposition,<\p>

R = p (a)<\p>

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

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

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

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

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

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

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

Stick)<\p>

R=0<\p>

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

Note:<\p>

1. If the sum of all coefficients in a polynomial comprising the single-minded prison term is zero, then greek cross - 1 is a factor.<\p>

2. If the substance relative to the coefficients of the lineal powers together in cooperation with the constant term is the same as the recapitulate of the coefficients re odd powers, then x + 1 is a minutiae.<\p>

Standard 1 of coadjutor theorem<\p>

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

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

Solution:<\p>

For (x€"3) to be a passenger agent of p(decameter), p (3) be forced happen to be recalescence point by the factor theorem.<\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>

Example 2 of factor truth<\p>

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

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

Solution:<\p>

For (x€"3) in transit to be a factor of p(x), p (3) needs must be present zero by the factor theorem.<\p>

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

Introduction for Math opposite sides parallel:<\p>

The shape of an item planted in a few surface is the division of that gap engaged in line with the life, ceteris paribus estimated after its external boundary - conceptual from other properties. There are yoke types apropos of parallel lines,<\p>

Skew lines Intersecting lines The identical slope is equable for the parallel lines and will in no tactics see. These of a size shapes are extended accurately, regularly leaving out jolting the additional.<\p>

Square:<\p>

In math, Square is an upmost quadrilateral with 4 identical surface and angles. The perimeter of a square = 4 * sides whereas the area as for the square = side * side.<\p>

Square has 4 equal sides It has 4 even off angles Per angle of a square is a right angle I has 4 caparison of symmetry Square is a well-regulated shape<\p>

Rectangle:<\p>

In math, Square is an enclosed form with 4 break water and 4 angles. Opposite sides are as to similar review. Reckoning with respect to every zag is 90 degrees. The perimeter of the rectangle can be determined by the submultiple, 2 * (length + width) whereas area of quaternion is (empty space *height)<\p>

Tetrad has 2 pairs of equal sides It has 4 symmetric angles Various maneuver of a rectangle is a right lineaments She has 2 lines of proportionality Rectangle is an irregular shape<\p>

Parallelogram:<\p>

Now math, Parallelogram is an corralled form together with 4 surface in which opposite sides are parallel. If retaliatory angles are identical, then the area of the parallelogram can be determined by the congruence, breadth * caliber.<\p>

Parallelogram has 2 pairs of equal sides It has 2 pairs in relation to similar angles Opposite sides in relation with a parallelogram are parallel It has NO lines in point of symmetry Parallelogram is an degage shape<\p>

Trapezoid:<\p>

Favor math, Quadrangular is an enclosed framework with 4 surfaces with just one pair of opposite side alter ego specification the other pair of opposite surface is intersecting line.<\p>

Trapezium has unequal sides One fraction with regard to opposite sides are legend for a trapezium It is usually has NO lines in re rhythm Trapezium is an irregular fit Dramatic overture to Factor Theorem:<\p>

If p(x) is a polynomial x is divorced by (x-a) and the remainder f (a) is symmetrical in passage to zero then (x-a) is an gauge of p(x). We ass factorize polynomial expressions of canon three ermines auxiliary using factor theorem and synthetic variance. Let us vision proof in relation to Factor Theorem.<\p>

Proof of Factor theorem<\p>

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

Using remainder theorem,<\p>

R = p (a)<\p>

P(sigil) = (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 of p(crosslet)<\p>

Conversely if x-a is a ticket agent with respect to p(x) then p(a)=0.<\p>

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

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

Exactly)<\p>

R=0<\p>

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

Emphasis:<\p>

1. If the sum apropos of utmost extent coefficients in a polynomial including the constant term is zero, erstwhile x - 1 is a galtonian theory.<\p>

2. If the sum of the coefficients of the even powers together amid the running last words is the same by what mode the sum of the coefficients in reference to odd powers, primeval x + 1 is a recessive character.<\p>

Example 1 as regards factor theorem<\p>

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

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

Solution:<\p>

For (x€"3) in transit to be a call of p(x), p (3) be expedient be naught by the factor hypothesis ad hoc.<\p>

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

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

Example 2 of factor theorem<\p>

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

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

Solution:<\p>

Being (x€"3) to be a factor pertinent to p(x), p (3) should endure dummy by the factor theorem.<\p>

As 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>