polynomial-160521065801

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Polynomials:

Polynomials

Followings are not Polynomial:

Followings are not Polynomial  

3x4 + 5x2 – 7x + 1 The polynomial above is in standard form. Standard form of a polynomial - means that the degrees of its monomial terms decrease from left to right.:

3x 4 + 5x 2 – 7x + 1 The polynomial above is in standard form. Standard form of a polynomial - means that the degrees of its monomial terms decrease from left to right.

State whether each expression is a polynomial. If it is, identify it.:

State whether each expression is a polynomial. If it is, identify it. 1) 7y - 3x + 4 trinomial 2) 10x 3 yz 2 monomial 3) not a polynomial

Find the degree of x5 – x3y2 + 4:

Find the degree of x 5 – x 3 y 2 + 4 0 2 3 5 10

3) Put in ascending order in terms of y: 12x2y3 - 6x3y2 + 3y - 2x:

3 ) Put in ascending order in terms of y : 12x 2 y 3 - 6x 3 y 2 + 3y - 2x -2x + 3y - 6x 3 y 2 + 12x 2 y 3 Put in ascending order: 5a 3 - 3 + 2a - a 2 -3 + 2a - a 2 + 5a 3

Write in ascending order in terms of y: x4 – x3y2 + 4xy – 2x2y3:

Write in ascending order in terms of y: x 4 – x 3 y 2 + 4xy – 2x 2 y 3 x 4 + 4xy – x 3 y 2 – 2x 2 y 3 – 2x 2 y 3 – x 3 y 2 + 4xy + x 4 x 4 – x 3 y 2 – 2x 2 y 3 + 4xy 4xy – 2x 2 y 3 – x 3 y 2 + x 4

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Remainder Theoram If p(x) a polynomial divided by a divisor g(x) of order 1(ex. x-a) And r(x) be the remainder then we can write P(x)=g(x).q(x) +r(x) P(x)= (x-a).q(x) + r(x) P(a)=r(a) Since g(x) is order of 1 hence r(x) will be order of zero(N.B: the order of remainder is always less than divisor.) Since the order of r(x) is zero hence it is constant. Suppose r(a)= r Therefore p(a)= r

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Q. On dividing the polynomial 4x² - 5x² - 39x² - 46x – 2 by the polynomial g(x) the quotient is x² - 3x – 5 and the remainder is -5x + 8.Find the polynomial g(x). Ans: p(x) = g(x).q(x)+r(x) let p(x) = 4x4 – 5x³ – 39x²– 46x – 2 q(x) = x²– 3x – 5 and r (x) = -5x + 8 now p(x) – r(x) = g(x).q(x) 4x² - 5x² - 39x² - 46x – 2 –(-5x+8)= (x² - 3x – 5 ).g(x) ∴ g(x) = 4x2 + 7x + 2

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Q. If the squared difference of the zeros of the quadratic polynomial x² + px + 45 is equal to 144 , find the value of p. Ans: Let two zeros are α and β where α > β According given condition ( α - β)2 = 144 Let p(x) = x² + px + 45 α + β = − p αβ = 45 now ( α + β)² = (α - β)² + 4 αβ (-P)² = 144 +180=324 Solving this we get p = ± 18

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Q. If α & ß are the zeroes of the polynomial 2x² ─ 4x + 5, then find the value of a. α² + ß² b. 1/ α + 1/ ß c. (α ─ ß)² d. 1/α² + 1/ß² e. α² + ß² Q. Obtain all the zeros of the polynomial p(x) = 3x4 ─ 15x³ + 17x² +5x ─6 if two zeroes are ─1/√3 and 1/√3 Q. If one zero of the polynomial 3x² - 8x +2k+1 is seven times the other, find the zeros and the value of k Q.If two zeros of the polynomial f(x) = x4 - 6x3 - 26x2 + 138x – 35 are 2±√3.Find the other zeros.

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Q. Find all zeroes of f(x) = 3x4+6x³-2x²-10x-5, if two of its zeroes are √ 5/3 and -√5/3. Since √ 5/3 and -√5/3. are zeroes of f(x). Therefore (x- √ 5/3 )(x+-√5/3) = (x²- 5/3) is a factor of f(x). After dividing f(x) by (x²- 5/3), we get

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As (x²+2x+1) = ( x+1)² Therefore for obtaining other zeroes of f(x) ,we put (x+1)² = 0 , Therefore x= -1 and -1 Hence zeroes of f(x) is are √ 5/3 and -√5/3. 1 and -1.

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Alternative Method: Say the other roots are α and β Therefore √ 5/3 -√5/3+α +β=-6/3=-2 α +β=-2……………….(i) Again√ 5/3 (-√5/3).α.β=-5/3 α.β=5/3 Using the algebraic identity α -β=………….… (ii) Solving (i) and (ii) we can find the other two roots

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Find zeros third degree polynomial equation, x³ - 2x² + 2x - 1 = 0 Let P(x) = , x³ - 2x² + 2x – 1 P(1) = (1) 3 - 2(1) 2 + 2(1) - 1 = 0 Since P(1) = 0, x - 1 is the factor of P(x). Long Division Method: x 3 - 2x 2 + 2x - 1 = (x - 1)(x 2 + 6x + 5 ) x 2 + 6x + 5 = x 2 + 5x + x + 5 = x(x + 5) + (x + 5) = (x + 1)(x + 5 ) x 3 - 2x 2 + 2x - 1 = (x - 1)(x + 1)(x + 5) = 0 Hence Zeros of the cubic equation are 1, -1, -5.

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Question : If two zeroes of the polynomial are find other zeros = x² + 4 − 4x − 3 = x² − 4x + 1 is a factor of the given polynomial For finding the remaining zeroes of the given polynomial, we will find the quotient by dividing

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= = Or x = 7 or −5 Hence, 7 and −5 are also zeroes of this polynomial.

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