If two zeroes of the polynomial x 4 + x 3 - 15 x 2 - 29 x - 6 are 2 ± 5 . Find the other zeroes.

Given polynomial is x 4 + x 3 - 15 x 2 - 29 x - 6 and two of its zeroes are 2 + 5 and 2 - 5 . Here, we can write x - ( 2 + 5 ) and x - ( 2 - 5 ) are factors of the polynomial. Then, x - ( 2 + 5 ) x - ( 2 - 5 ) = ( x 2 - 4 x - 1 ) is also the factor of polynomial. Now, we divide x 4 + x 3 - 15 x 2 - 29 x - 6 by ( x 2 - 4 x - 1 ) as shown below: x 2 - 4 x - 1 ) x 4 + x 3 - 15 x 2 - 29 x - 6 ( x 2 + 5 x + 6 x 4 - 4 x 3 - x 2 - + + 0 + 5 x 3 - 14 x 2 - 29 x - 6 + 5 x 3 - 20 x 2 - 5 x - + + 0 + 6 x 2 - 24 x - 6 + 6 x 2 - 24 x - 6 - + + 0 Using the division algorithm, we can write, x 4 + x 3 - 15 x 2 - 29 x - 6 = ( x 2 + 5 x + 6 ) ( x 2 - 4 x - 1 ) + 0 = [ x 2 + 3 x + 2 x + 6 ] ( x 2 - 4 x - 1 ) = [ x ( x + 3 ) + 2 ( x + 3 ) ] ( x 2 - 4 x - 1 ) = ( x + 2 ) ( x + 3 ) ( x 2 - 4 x - 1 ) To find the remaining zeroes, we equate ( x + 2 ) ( x + 3 ) to 0 . ( x + 2 ) ( x + 3 ) = 0 ⇒ x + 2 = 0 , x + 3 = 0 ⇒ x = - 2 , - 3 Therefore, the remaining zeroes are - 2 , - 3 .

Given that 2 is a zero of a cubic polynomial 6 x 3 + 2 x 2 - 10 x - 4 2 , find its other two zeroes.

Given polynomial is p x = 6 x 3 + 2 x 2 - 10 x - 4 2 and one of its zeroes is 2 . Here, we can write x - 2 is a factor of the polynomial. Now, divide the polynomial p x by ( x - 2 ) as shown below: x - 2 ) 6 x 3 + 2 x 2 - 10 x - 4 2 ( 6 x 2 + 7 2 x + 4 6 x 3 - 6 2 x 2 - + 0 + 7 2 x 2 - 10 x - 4 2 + 7 2 x 2 - 14 x + + 0 + 4 x - 4 2 4 x - 4 2 - + 0 Using the division algorithm, we can write, 6 x 3 + 2 x 2 - 10 x - 4 2 = ( 6 x 2 + 7 2 x + 4 ) × ( x - 2 ) + 0 = [ 6 x 2 + 4 2 x + 3 2 x + 4 ] × ( x - 2 ) = [ 2 x ( 3 x + 2 2 ) + 2 ( 3 x + 2 2 ) ] ( x - 2 ) = ( 2 x + 2 ) ( 3 x + 2 2 ) ( x - 2 ) Finding the other zeroes, ( 2 x + 2 ) ( 3 x + 2 2 ) = 0 ⇒ 2 x + 2 = 0 or 3 x + 2 2 = 0 ⇒ x = - 2 2 or x = - 2 2 3 . Therefore, - 2 2 , - 2 2 3 are the remaining zeroes of polynomial.

Given that x - 5 is a factor of the cubic polynomial x 3 - 3 5 x 2 + 13 x - 3 5 . Find all the zeroes of the polynomial.

Given: x - 5 is a factor of the cubic polynomial x 3 - 3 5 x 2 + 13 x - 3 5 . Dividing x 3 - 3 5 x 2 + 13 x - 3 5 by x - 5 . x - 5 ) x 3 - 3 5 x 2 + 13 x - 3 5 ( x 2 - 2 5 x + 3 x 3 - 5 x 2 - + 0 - 2 5 x 2 + 13 x - 3 5 - 2 5 x 2 + 10 x + - 0 + 3 x - 3 5 + 3 x - 3 5 - + 0 Using the division algorithm, we can write, x 3 - 3 5 x 2 + 13 x - 3 5 = ( x 2 - 2 5 x + 3 ) ( x - 5 ) + 0 To find the other zeroes, we find the zeroes of x 2 - 2 5 x + 3 . a = 1 , b = - 2 5 , c = 3 x = - b ± b 2 - 4 a c 2 a = 2 5 ± ( - 2 5 ) 2 - 4 ( 1 ) ( 3 ) 2 ( 1 ) = 2 5 ± 20 - 12 2 = 2 5 ± 8 2 = 2 5 ± 2 2 2 = 5 ± 2 Hence, all the zeroes of the polynomial are 5 , 5 + 2 , 5 - 2 .

E M B I B E

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE

H K Dass, Rama Verma and, Bhagwat Swarup Sharma Solutions for Chapter: Polynomials, Exercise 4: EXERCISE

Author:H K Dass, Rama Verma & Bhagwat Swarup Sharma

H K Dass Mathematics Solutions for Exercise - H K Dass, Rama Verma and, Bhagwat Swarup Sharma Solutions for Chapter: Polynomials, Exercise 4: EXERCISE

Attempt the free practice questions on Chapter 2: Polynomials, Exercise 4: EXERCISE with hints and solutions to strengthen your understanding. New Mathematics for Class X solutions are prepared by Experienced Embibe Experts.

Questions from H K Dass, Rama Verma and, Bhagwat Swarup Sharma Solutions for Chapter: Polynomials, Exercise 4: EXERCISE with Hints & Solutions

EASY

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE>Q 2

Check whether the polynomial $g (x)$ is a factor of the polynomial $p (x)$ by division algorithm, in the following cases.

$p (x) = 9 x^{4} + 4 x^{3} - 2 x^{2} + x + 10; g (x) = 3 x^{2} + 2 x + 5$

EASY

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE>Q 2

Check whether the polynomial $g (x)$ is a factor of the polynomial $p (x)$ by division algorithm, in the following cases.

$p (x) = 8 x^{4} + 6 x^{3} - 3 x^{2} + x + 6; g (x) = 4 x^{2} + 2 x + 3$

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE>Q 3

Obtain all zeroes of $p (x) = x^{4} - 3 x^{3} - x^{2} + 9 x - 6$ if two of its zeroes are $(- \sqrt{3})$ and $\sqrt{3}$ .

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE>Q 4

If two zeroes of the polynomial $x^{4} + x^{3} - 15 x^{2} - 29 x - 6$ are $2 \pm \sqrt{5}$ . Find the other zeroes.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE>Q 5

Given that $\sqrt{2}$ is a zero of a cubic polynomial $6 x^{3} + \sqrt{2} x^{2} - 10 x - 4 \sqrt{2}$ , find its other two zeroes.

HARD

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE>Q 6

Find $k$ so that $x^{2} + 2 x + k$ is a factor $2 x^{4} + x^{3} - 14 x^{2} + 5 x + 6$ . Also, find all the zeroes of the two polynomials.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE>Q 7

Given that $x - \sqrt{5}$ is a factor of the cubic polynomial $x^{3} - 3 \sqrt{5} x^{2} + 13 x - 3 \sqrt{5}$ . Find all the zeroes of the polynomial.

MEDIUM

10th CBSE

IMPORTANT

New Mathematics for Class X>Chapter 2 - Polynomials>EXERCISE>Q 8

For which values of $a$ and $b$ , are zeroes of $q (x) = x^{3} + 2 x^{2} + a$ also the zeroes of the polynomial $p (x) = x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x + b$ ? Which zeroes of $p (x)$ are not the zeroes of $q (x)$ ?

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