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 zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
4s2 – 4s + 1
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate α - β
If α and β are the zeros of the quadratic polynomial f(x) = ax2 + bx + c, then evaluate `1/alpha+1/beta-2alphabeta`
If α and β are the zeros of the quadratic polynomial f(x) = 6x2 + x − 2, find the value of `alpha/beta+beta/alpha`.
Find the zeroes of the quadratic polynomial `2x^2 ˗ 11x + 15` and verify the relation between the zeroes and the coefficients.
Find the quadratic polynomial, sum of whose zeroes is 0 and their product is -1. Hence, find the zeroes of the polynomial.
If `x =2/3` and x = -3 are the roots of the quadratic equation `ax^2+2ax+5x ` then find the value of a and b.
What should be subtracted to the polynomial x2 − 16x + 30, so that 15 is the zero of the resulting polynomial?
Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),
where p(x) = x5 − 4x3 + x2 + 3x +1, g(x) = x3 − 3x + 1
An asana is a body posture, originally and still a general term for a sitting meditation pose, and later extended in hatha yoga and modern yoga as exercise, to any type of pose or position, adding reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that poses can be related to representation of quadratic polynomial.
The zeroes of the quadratic polynomial `4sqrt3"x"^2 + 5"x" - 2sqrt3` are:
Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the product of the other two zeroes is ______.
If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:
`4x^2 + 5sqrt(2)x - 3`
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
`(-8)/3, 4/3`
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
`-2sqrt(3), -9`
If one zero of the polynomial p(x) = 6x2 + 37x – (k – 2) is reciprocal of the other, then find the value of k.
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 the sum and product of whose zeroes are – 3 and 2 respectively, is ______.
The zeroes of the polynomial p(x) = 2x2 – x – 3 are ______.
Find the zeroes of the quadratic polynomial 4s2 – 4s + 1 and verify the relationship between the zeroes and the coefficients.
