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^{th}Polynomials Ch-02-Mathematics [Key points]^{2}+ 3x + 2 ,y

^{2}+ 2

^{2}+ bx + c, where a, b, c are real numbers and a ≠ 0.

*x*

^{3},

*x*

^{3}, √2

*x*

^{3}.

^{3}+ bx

^{2}+ c x + d, where, a, b, c, d are real numbers and a ≠ 0.

*k*is a zero of

*p*(

*x*) =

*ax*+

*b*, then

*p*(

*k*) = a k + b = 0, i.e., k = -b/a

**parabolas**.

^{2}+ bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola representing y = ax

^{2}+ bx + c intersects the x-axis.

^{2}+ b x + c, a ≠ 0, then you know that x – α and x – β are the factors of p(x).

Therefore, ax

^{2}+ bx + c = a(x – α) (x – β)= ax

^{2}– a(α + β)x + a α β]

^{2}, x and constant terms on both the sides, we get , a = k

**,**b = – k(α + β) and c = kαβ.

**α + β = -b/a ; α β = c/a**

^{3}+ bx

^{2}+ c x + d= a(x-a)(x-b)(x-g) = ax

^{3}– a(a+b+g)x

^{2 }+ a(ab +bg+ga) - aabg

Þ Division algorithm states that given any polynomial

*p*(

*x*) and any non-zero polynomial

*g*(

*x*), there are polynomials

*q*(

*x*) and

*r*(

*x*) such that

*p*(

*x*) =

*g*(

*x*)

*q*(

*x*) +

*r*(

*x*), where

*r*(

*x*) = 0 or degree

*r*(

*x*) < degree

*g*(

*x*).

^{2}+ 7x + 10, and verify the relationship between the zeroes and the coefficients.

^{2}– 3 and verify the relationship between the zeroes and the coefficients.

^{3 }– 5x

^{2}– 11x – 3, and then verify the relationship between the zeroes and the coefficients.

^{4 }– 3x

^{3}– 3x

^{2}+ 6x – 2, if you know that two of its zeroes are √2 and − √2

^{4}+ 6x

^{3}– 2x

^{2}– 10x – 5, if two of its zeroes are √(5/3) , -√(5/3)

*x*

^{3}– 3

*x*

^{2}+

*x*+ 1 are

*a*–

*b*,

*a*,

*a*+

*b*, find

*a*and

*b*.

**If two zeroes of the polynomial**

*x*

^{4}– 6

*x*

^{3}– 26

*x*

^{2}+ 138

*x*– 35 are 2 ± √3 , find other zeroes.

**If the polynomial**

*x*

^{4}– 6

*x*

^{3}+ 16

*x*

^{2}– 25

*x*+ 10 is divided by another polynomial

*x*

^{2}– 2

*x*+

*k*, the remainder comes out to be

*x*+

*a*, find

*k*and

*a*.

*f(x) = x*, Show that (α+ 1) (β + 1) = 1- c.

^{2}- p (x+1) - cQ.13. 1. For which values of a and b , are the zeros of g(x) = x

^{3}+2x

^{2}+a, also the zeros of the polynomial f(x) =x

^{5}-x

^{4}-4x

^{3}+3x

^{2}+3x+b ? Which zeros of f(x) are not the zeros of g(x)?

(Ans. 1 and 2 are the zeros of g(x) which are not the zeros f(x) and this happens when a= -2 , b= -2)

^{2}- 8x –( 2k + 1) is seven times the other , find both zeroes of the polynomial and the value of k .

^{2}+2ax+5x+10then find the value of a. [Ans : 2]

^{4}– 6x

^{3}– 26x

^{2}+138x -35 are 2- Root3 and 2+ root3 , find all the zeros .

**Given**system of equations is:

Ans: Let the number of questions correctly answered be x

Therefore, number of questions incorrectly answered will be 120 - x.

According to the given condition,

1× x – (1/2)(120-x) = 90

⇒ x - 60 +(1/2)( x = 90

⇒ (3/2) x = 150

Class X Polynomial Test Paper-1

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