A day full of math games & activities. Find one near you.

Given that the zeroes of the cubic polynomial x³ - 6x² + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial

Solution:

Given, the cubic polynomial is x³ - 6x² + 3x + 10.

The zeros are of the form a, a + b, a + 2b.

We have to find the values of a and b and the zeros of the given polynomial.

We know that, if 𝛼, ꞵ and 𝛾 are the zeroes of a cubic polynomial ax³ + bx² + cx + d, then

𝛼 + ꞵ + 𝛾 = -b/a = -coefficient of x²/coefficient of x³

𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = c/a = coefficient of x/coefficient of x³

𝛼ꞵ𝛾 = -d/a = -constant/coefficient of x³

Here, 𝛼 = a, ꞵ = a + b and 𝛾 = a + 2b.

coefficient of x³ term, a = 1

coefficient of x² term, b = -6

coefficient of x term, c = 3

coefficient of constant term, d = 10

Sum of the roots:

𝛼 + ꞵ + 𝛾 = a + a + b + a + 2b

= 3a + 3b

= 3(a + b)

-b/a = -(-6)/1

= 6

3(a + b) = 6

So, a + b = 2

a = 2 - b ----------- (1)

Product of all the roots:

𝛼ꞵ𝛾 = a.(a+b)(a+2b)

Put the value of a,

= (2 - b)(2 - b + b)(2 - b + 2b)

= (2 - b)(2)(2 + b)

-d/a = -10/1 = -10

(2 - b)(2)(2 + b) = -10

(2 - b)(2 + b) = -5

By using algebraic identity,

(a - b)(a + b) = a² - b²

4 - b² = -5

b² = 9

Taking square root,

b = ±3

When b = +3, a = 2 - 3 = -1

When b = -3, a = 2 - (-3) = 2 + 3 = 5

Finding the zeros,

When a =5, b = -3

a = 5

a + b = 5 - 3 = 2

a + 2b = 5 + 2(-3) = 5 - 6 = -1

Therefore, the zeros are 5, 2, -1.

When a = -1, b = 3

a = -1

a + b = -1 + 3 = 2

a + 2b = -1 + 2(3) = -1 + 6 = 5.

Therefore, the zeros are -1, 2 and 5.

✦ Try This: Given that the zeroes of the cubic polynomial 2x³ - x² + 3x + 9 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as

the zeroes of the given polynomial

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 2

NCERT Exemplar Class 10 Maths Exercise 2.4 Problem 2

Given that the zeroes of the cubic polynomial x³ - 6x² + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial

Summary:

Given that the zeroes of the cubic polynomial x³ - 6x² + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, the values of a and b are -1 or 5 and -3 or +3. The zeroes of the given polynomial are -1, 2 and 5.

☛ Related Questions: