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For which values of a and b, are the zeroes of q(x) = x³ + 2x² + a also the zeroes of the polynomial p(x) = x⁵ - x⁴ - 4x³ + 3x² + 3x + b? Which zeroes of p(x) are not the zeroes of q(x)

Solution:

Given, p(x) = x⁵ - x⁴ - 4x³ + 3x² + 3x + b

g(x) = x³ + 2x² + a

We have to determine which zeros of p(x) are not the zeros of q(x).

The 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).

For the zeros of q(x) to the zeros of p(x), q(x) must be the factor of p(x).

This implies that the remainder is zero.

By using long division,

For which values of a and b, are the zeroes of q(x) = x³ + 2x² + a also the zeroes of the polynomial p(x) = x⁵ - x⁴ - 4x³ + 3x² + 3x + b? Which zeroes of p(x) are not the zeroes of q(x)

So, (x⁵ - x⁴ - 4x³ + 3x² + 3x + b)/ (x³ + 2x² + a) = (x² - 3x + 2) + x²(-1-a) + x(3+3a) + b - 2a

As remainder is zero, x²(-1-a) + (3+3a)x + b - 2a = 0

Coefficient of x² = (-1-a) = 0

-1 = a

a = -1

Coefficient of x = 3 + 3a = 0

3 = -3a

a = -3/3

a = -1

Constant = b - 2a = 0

b = 2a

b = 2(-1)

b = -2

Therefore, the values of a and b are -1 and -2.

Now, p(x) = x⁵ - x⁴ - 4x³ + 3x² + 3x - 2

q(x) = x³ + 2x² - 1

Other zeros of q(x) can be found by factoring,

x² - 3x + 2 = 0

x² - x - 2x + 2 = 0

x(x - 1) - 2(x - 1) = 0

(x - 2)(x - 1) = 0

Now, x - 2 = 0

x = 2

Also, x - 1 = 0

x = 1

Therefore, other zeros of p(x) are 1 and 2.

✦ Try This: For which values of a and b, are the zeroes of q(x) = 2x³ + 4x² + a also the zeroes

of the polynomial p(x) = 2x⁵ - 3x⁴ - 4x³ + 2x² + x + b? Which zeroes of p(x) are

not the zeroes of q(x)

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

NCERT Exemplar Class 10 Maths Exercise 2.4 Problem 6

For which values of a and b, are the zeroes of q(x) = x³ + 2x² + a also the zeroes of the polynomial p(x) = x⁵ - x⁴ - 4x³ + 3x² + 3x + b? Which zeroes of p(x) are not the zeroes of q(x)

Summary:

For values of a and b, the zeroes of q(x) = x³ + 2x² + a is also the zeroes of the polynomial p(x) = x⁵ - x⁴ - 4x³ + 3x² + 3x + b is a = -1 and b = -2. The zeroes of p(x) which are not the zeroes of q(x) are 1 and 2

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