Advertisements
Advertisements
Question
Verify that the numbers given along side of the cubic polynomials are their zeroes. Also verify the relationship between the zeroes and the coefficients.
`2x^3 + x^2 – 5x + 2 ; 1/2, 1, – 2`
Advertisements
Solution
p(x) = 2x3 + x2 - 5x + 2
Zeroes for this polynomial are 1/2, 1, -2
p(1/2) = `2(1/2)^3+(1/2)^2-5(1/2)+2`
= `1/4+1/4-5/2+2`
=0
p(1) = 2 x 13 + 12 - 5 x 1 + 2
= 0
p(-2) = 2(-2)3 + (-2)2 - 5(-2) + 2
= -16 + 4 + 10 + 2 = 0
Therefore, 1/2, 1, and −2 are the zeroes of the given polynomial.
Comparing the given polynomial with ax3 + bx2 + cx + d we obtain a = 2, b = 1, c = −5, d = 2
We can take α = 1/2, β = 1, y = -2
α + β + γ = `1/2+1+(-2) = -1/2 = (-b)/a`
αβ + βγ + αγ = `1/2xx1+1(-2)+1/2(-2)=(-5)/2 = c/a`
αβγ = `1/2xx1xx(-2) = (-1)/1=(-(2))/2=(-d)/a`
Therefore, the relationship between the zeroes and the coefficients is verified.
APPEARS IN
RELATED QUESTIONS
Find the zeros of the quadratic polynomial 4x2 - 9 and verify the relation between the zeros and its coffiecents.
if α and β are the zeros of ax2 + bx + c, a ≠ 0 then verify the relation between zeros and its cofficients
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients.
4u2 + 8u
Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficients:
3x2 – x – 4
Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.
1, 1
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 α and β are the zeros of the quadratic polynomial f(x) = 6x2 + x − 2, find the value of `alpha/beta+beta/alpha`.
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 𝛼 and 𝛽 are the zeros of the quadratic polynomial p(x) = 4x2 − 5x −1, find the value of α2β + αβ2.
By actual division, show that x2 – 3 is a factor of` 2x^4 + 3x^3 – 2x^2 – 9x – 12.`
If 2 and -2 are two zeroes of the polynomial `(x^4 + x^3 – 34x^2 – 4x + 120)`, find all the zeroes of the given polynomial.
If two zeros x3 + x2 − 5x − 5 are \[\sqrt{5}\ \text{and} - \sqrt{5}\], then its third zero is
What should be added to the polynomial x2 − 5x + 4, so that 3 is the zero of the resulting polynomial?
If \[\sqrt{5}\ \text{and} - \sqrt{5}\] are two zeroes of the polynomial x3 + 3x2 − 5x − 15, then its third zero is
The polynomial which when divided by −x2 + x − 1 gives a quotient x − 2 and remainder 3, 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
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.
Find the zeroes of the polynomial x2 + 4x – 12.
The zeroes of the polynomial p(x) = 25x2 – 49 are ______.
