#factor theorem

Unit 21: Poly Functions Zeros

Part 1 —>

Remainder Theorem

#1

Factor Theorem

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Rational Zero Theorem

#1

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>

O Level Additional Mathematics – Remainder and Factor Theorems

Notes 1. Remainder Theorem If a polynomial f(x) is divided by x – a, then the remainder, R = f(a). 2. Factor Theorem If f(a) = 0, then (x – a) is a factor of f(x) 3. Solutions of Equations To solve the equation f(x) = 0, first factorize f(x) by the Factor Theorem Eg . solve x3 – 2x2 – 5x + 6 = 0 Let f(x) = x3 – 2x2 – 5x + 6 Find a factor by trial and error eg, found that (x – 1) is a factor…

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>