Frequently Asked Questions on Chapter 2- Polynomials

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?

Graphical method to find zeroes:- Total number of zeroes in any polynomial equation = total number of times the curve intersects x-axis. In the given graph, the number of zeroes of p(x) is 0 because the graph is parallel to x-axis does not cut it at any point.

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients 4u²+ 8u?

4u(u+2) Therefore, zeroes of polynomial equation 4 + 8u are {0, -2}. Sum of zeroes = 0+(-2) = -2 =-8/4 =(-coefficient of u)/coefficient of u²
Product of zeroes = 0x-2 = 0 = 0/4 =constant term/coefficient of u²

Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and deg p(x) = deg q(x)?

According to the division algorithm, dividend p(x) and divisor g(x) are two polynomials, where g(x)≠0. Then we can find the value of quotient q(x) and remainder r(x), with the help of below given formula; Dividend = Divisor × Quotient + Remainder ∴ p(x) = g(x) × q(x) + r(x) Where r(x) = 0 or degree of r(x)< degree of g(x). Now let us proof the three given cases as per division algorithm by taking examples for each.
deg p(x) = deg q(x)
Degree of dividend is equal to degree of quotient, only when the divisor is a constant term.Let us take an example, 3x²+3x+3 is a polynomial to be divided by 3. So, 3x²+3x+3/3=x²+x+1=q(x) Thus, you can see, the degree of quotient is equal to the degree of dividend. Hence, division algorithm is satisfied here.

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?

Let us consider the cubic polynomial is ax³+bx²+cx+d and the values of the zeroes of the polynomials be α, β, γ. As per the given question, α + β + γ = -b/a = 2/1 αβ + βγ + γα = c/a = -7/1 α β γ = -d/a = -14/1 Thus, from above three expressions we get the values of coefficient of polynomial. a = 1, b = -2, c = -7, d = 14 Hence, the cubic polynomial is x³-2x²-7x+14

Find the zeros in the given x^2 – 2x – 8 quadratic polynomial and verify the relationship between the zeroes and the coefficients.

Let us simplify the given quadratic polynomial,

x^2 – 2x – 8 = x^2 – 4x + 2x – 8

= x(x – 4) + 2(x – 4)

= (x – 4) (x + 2)

Therefore, zeros of polynomial equation x^2 – 2x – 8 are (4, -2)

So,

Sum of zeros = 4 – 2 = 2 = -(-2)/1 = -(Coefficient of x) / (Coefficient of x^2)

Product of zeros = 4×(-2) = -8 = -(8)/1 = (Constant term) / (Coefficient of x^2)

Find a quadratic polynomial with the given numbers 1/4 , -1 as the sum and product of its zeros respectively.

By using the formulas,

Sum of zeros = α + β

Product of zeros = α β

So,

Sum of zeros = α + β = 1/4

Product of zeros = α β = -1

Therefore, If α and β are zeros of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:

x^2 – (α + β)x + αβ = 0

x^2 – (1/4)x + (-1) = 0

4x^2 – x – 4 = 0

Hence, 4x^2 – x – 4 is the quadratic polynomial.

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 2, –7, –14 respectively.

Let us consider the cubic polynomial is ax^3 + bx^2 + cx + d and the values of the zeroes of the polynomials are α, β, γ.

So according to the question,

α + β + γ = -b/a = 2/1

αβ + βγ + γα = c/a = -7/1

αβγ = -d/a = -14/1

Thus, from the above three expressions we get the values of coefficient of polynomial.

a = 1, b = -2, c = -7, d = 14

Hence, the cubic polynomial is x^3 – 2x^2 – 7x + 14