Advertisements
Advertisements
Question
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
x3 – 4x2 + 5x – 2; 2, 1, 1
Advertisements
Solution
x3 – 4x2 + 5x – 2; 2, 1, 1
p(x) = x3 − 4x2 + 5x − 2 .... (1)
Zeroes for this polynomial are 2,1,1
Substitute x=2 in equation (1)
p(2) = 23 − 4 × 22 + 5 × 2 − 2
= 8 − 16 + 10 − 2 = 0
Substitute x=1 in equation (1)
p(1) = x3 − 4x2 + 5x − 2
= 13 − 4(1)2 + 5(1) − 2
= 1 − 4 + 5 − 2 = 0
Therefore, 2,1,1 are the zeroes of the given polynomial.
Comparing the given polynomial with ax3 + bx2 + cx + d we obtain,
a = 1, b = −4, c = 5, d = −2
Let us assume α = 2, β = 1, γ = 1
Sum of the roots = α + β + γ = 2 + 1 + 1 = 4 = `- (-4)/1 (-"b")/"a"`
Multiplication of two zeroes taking two at a time = αβ + βγ + αγ = (2)(1) + (1)(1) + (2)(1) = 5 = `5/1 = "c"/"a"`
Product of the roots = αβγ = 2 × 1 × 1 = 2 = `−(-2)/1="d"/"a"`
APPEARS IN
RELATED QUESTIONS
Find the zeros of the quadratic polynomial 4x2 - 9 and verify the relation between the zeros and its coffiecents.
Prove relation between the zeros and the coefficient of the quadratic polynomial ax2 + bx + c
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
`q(x)=sqrt3x^2+10x+7sqrt3`
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `beta/(aalpha+b)+alpha/(abeta+b)`
If a and are the zeros of the quadratic polynomial f(x) = 𝑥2 − 𝑥 − 4, find the value of `1/alpha+1/beta-alphabeta`
If one zero of the quadratic polynomial f(x) = 4x2 − 8kx − 9 is negative of the other, find the value of k.
If the zeros of the polynomial f(x) = 2x3 − 15x2 + 37x − 30 are in A.P., find them.
Find the quadratic polynomial whose zeroes are `2/3` and `-1/4` Verify the relation between the coefficients and the zeroes of the polynomial.
If f(x) =` x^4 – 3x^2 + 4x + 5` is divided by g(x)= `x^2 – x + 1`
If 1 and –2 are two zeroes of the polynomial `(x^3 – 4x^2 – 7x + 10)`, find its third zero.
If 𝛼, 𝛽 are the zeroes of the polynomial f(x) = x2 + x – 2, then `(∝/β-∝/β)`
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, is
Zeroes of a polynomial can be determined graphically. No. of zeroes of a polynomial is equal to no. of points where the graph of polynomial ______.
The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.




If the sum of the roots is –p and the product of the roots is `-1/"p"`, then the quadratic polynomial is:
The number of polynomials having zeroes as –2 and 5 is ______.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
t3 – 2t2 – 15t
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`2x^2 + (7/2)x + 3/4`
Given that `sqrt(2)` is a zero of the cubic polynomial `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`, find its other two zeroes.
If α, β are the zeroes of the polynomial p(x) = 4x2 – 3x – 7, then `(1/α + 1/β)` is equal to ______.
A quadratic polynomial whose sum and product of zeroes are 2 and – 1 respectively is ______.
