NCERT Class 10 Maths Solutions Chapter - 2 Polynomials, Ex-2.4

Ex - 2.4

Question 1. 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:


Solution



On comparing the given polynomial with the polynomial ax³ + bx² + cx + d, we obtain a = 2, b = 1, c = -5, d = 2



Thus, the relationship between the zeroes and the coefficients is verified.

On comparing the given polynomial with the polynomial ax_³ + bx_² + cx + d, we obtain a = 1, b = -4, c = 5, d = -2.


Thus, the relationship between the zeroes and the coefficients is verified.

Concept insight: The zero of a polynomial is that value of the variable which makes the polynomial 0. Remember that there are three  relationships between the zeroes of a cubic polynomial and its coefficients which involve the sum of zeroes, product of all zeroes and the product of zeroes taken two at a time.

Question 2.  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.

Solution

Let the polynomial be ax_³  + bx_² + cx + d and its zeroes be   .
According to the given information:

If a = k, then b = -2k, c = -7k, d = 14k

Thus, the required cubic polynomial is k(x_³ - 2x_² - 7x + 14), where k is any real number.
The simplest polynomial will be obtained by taking k = 1.

Concept insight: A cubic polynomial involves four unknowns and we have three relations involving these unknowns, so the coefficients of the cubic polynomial can be found by assuming the value of one unknown and then finding the other three unknowns from the three relations.

Question 3.  If the zeroes of the polynomial x3 - 3x2 + x + 1 are a - b, a, a + b, find a and b.

Solution

Let p(x) = x³ - 3x² + x + 1
The zeroes of the polynomial p(x) are given as a - b, a, a + b.
Comparing the given polynomial with dx³ + ex² + fx + g, we can observe that
d = 1, e = -3, f = 1, g = 1



Therefore, the zeroes of p(x) are 1-b, 1, 1+b . 



Concept insight: When the polynomial and its zeroes are given then, remember to apply relationships between the zeroes and coefficients .
These relations  involve the sum of the zeroes, product of zeroes and the product of zeroes taken two at a time.

Question 4.  If two zeroes of the polynomial x4 - 6x3 - 26x2 + 138x - 35 are , find other zeroes.

Solution



Therefore, x_² - 4x + 1 is a factor of the given polynomial.

For finding the remaining zeroes of the given polynomial, we will carry out the division of

Hence, 7 and -5 are also zeroes of the given polynomial. 

Concept insight: If a is a zero of a polynomial p(x), where degree of p(x) is greater than 1, then (x - a) will be a factor of p(x) that is when p(x) is divided by (x - a), then the remainder obtained will be 0 and the quotient will be a factor of p(x) using division algorithm.  Remember that if (x - a) and (x - b) are factors of a polynomial, then (x - a)(x - b) will also be a factor of that polynomial.

Question 5.  If the polynomial x4 - 6x3 + 16x2 - 25x + 10 is divided by another polynomial x2 - 2x + k, the remainder comes out to be x + a, find k and a.

Solution

By division algorithm,
Dividend = Divisor x Quotient + Remainder
Divisor x Quotient = Dividend - Remainder

It will be perfectly divisible by x^₂ - 2x + k. 

10 - a - 8 x 5 + 25 = 0
10 - a - 40 + 25 = 0
- 5 - a = 0
a = -5

Thus, k = 5 and a = -5.

Concept insight: When a polynomial is divided by any non-zero polynomial, then it satisfies the division algorithm which is as below:
Dividend = Divisor x Quotient + Remainder

Divisor x Quotient = Dividend - Remainder 

So, from this relation, it can be said that the result (Dividend - Remainder) will be completely divisible by the divisor.