Welcome to the NCERT Solutions for Class 10th Mathematics - Chapter Polynomials. This page offers a step-by-step solution to the specific question from Exercise 3, Question 2: **check whether the first polynomial is a factor of...**.

Question 2

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

(i) t^{2} – 3, 2t^{4} + 3t^{3} – 2t^{2} – 9t – 12

(ii) x^{2} + 3x + 1, 3x^{4} + 5x^{3} – 7x^{2} + 2x + 2

(iii) x^{3} – 3x + 1, x^{5} – 4x^{3} + x^{2} + 3x + 1

Answer

**Note: If on dividing the second polynomial by first we get zero remainder then we say that first Is factor of second polynomial. **

(i) Given ,

First polynomial = t^{2}-3

Second polynomial = 2t^{4 }+3t^{3}-2t^{2 }-9t-12

As we can see the remainder is 0. Thereofre we can say that first polynomial is a factor of second polynomial.

(ii) Given,

First polynomial = x^{2}+3x+1

Second polynomial = 3x^{4 }+ 5x^{3 – }7x^{2 }+ 2x + 2

- Q:-
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x

^{2}– 2x – 8 (ii) 4s^{2}– 4s + 1 (iii) 6x^{2}– 3 – 7x (iv) 4u^{2}+ 8u (v) t^{2 }– 15 (vi) 3x^{2 }– x – 4 - Q:-
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

- Q:-
The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

- Q:-
On dividing x

^{3 }– 3x^{2 }+ x + 2 by a polynomial g(x), the quotient and remainder were x – 2 nd –2x + 4, respectively. Find g(x). - Q:-
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.

- Q:-
- Q:-
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

- Q:-
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :

(i) p(x) = x^{3 }– 3x^{2 }+ 5x – 3, g(x) = x^{2 }– 2 (ii) p(x) = x^{4 }– 3x^{2 }+ 4x + 5, g(x) = x^{2 }+ 1 – x (iii) p(x) = x^{4}– 5x + 6, g(x) = 2 – x^{2} - Q:-
If the polynomial x

^{4 }– 6x^{3 }+ 16x^{2 }– 25x + 10 is divided by another polynomial x^{2 }– 2x + k, the remainder comes out to be x + a, find k and a. - Q:-
Obtain all other zeroes of 3x

^{4}+ 6x^{3}– 2x^{2}– 10x – 5, if two of its zeroes are

- Q:-
Use Euclid’s division algorithm to find the HCF of :

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 - Q:-
A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30° (see Fig. 9.11).

- Q:-
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

- Q:-
Complete the following statements:

(i) Probability of an event E + Probability of the event ‘not E’ =

(ii) The probability of an event that cannot happen is

(iii) The probability of an event that is certain to happen is

(iv) The sum of the probabilities of all the elementary events of an experiment is(v) The probability of an event is greater than or equal to

- Q:-
Check whether the following are quadratic equations :

(i) (x + 1)^{2}= 2(x – 3) (ii) x^{2}– 2x = (–2) (3 – x) (iii) (x – 2)(x + 1) = (x – 1)(x + 3) (iv) (x – 3)(2x +1) = x(x + 5)(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) x

^{2}+ 3x + 1 = (x – 2)^{2}(vii) (x + 2)^{3}= 2x (x2 – 1) (viii) x^{3}– 4x^{2}– x + 1 = (x – 2)^{3} - Q:-
How many tangents can a circle have?

- Q:-
Show that any positive odd integer is of the form 6

*q*+ 1, or 6*q*+ 3, or 6*q*+ 5, where*q*is some integer. - Q:-
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

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The coach of a cricket team buys 3 bats and 6 balls for ` 3900. Later, she buys another bat and 3 more balls of the same kind for ` 1300. Represent this situation algebraically and geometrically.

- Q:-
Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.

(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

(iv) A baby is born. It is a boy or a girl.

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Prove that the parallelogram circumscribing a circle is a rhombus.

- Q:-
Find the roots of the following equations:

- Q:-
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x

^{2 }– 3x + 5 = 0 (iii) 2x^{2}– 6x + 3 = 0 - Q:-
A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8 (see Fig. 15.5 ), and these are equally likely outcomes. What is the probability that it will point at

(i) 8 ?

(ii) an odd number?

(iii) a number greater than 2?

(iv) a number less than 9? - Q:-
Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

- Q:-
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

- Q:-
Fill in the blanks :

(i) A tangent to a circle intersects it in

(ii) A line intersecting a circle in two points is called a

(iii) A circle can have

(iv) The common point of a tangent to a circle and the circle is called - Q:-
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m

^{2}? If so, find its length and breadth. - Q:-
One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting

(i) a king of red colour (ii) a face card (iii) a red face card

(iv) the jack of hearts (v) a spade (vi) the queen of diamonds - Q:-
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to

(A) 50° (B) 60°

(C) 70° (D) 80°

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