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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 the zeroes be α, β,γ.
Given sum of the zeros=α+β+γ = 2
Sum of the product of it's zeros taken two at a time = αβ+βγ+γα = -7
Also, product of it's zeros= αβγ= -14
The cubic polynomial is of the form
x³ - (sum of the zeros) x² +(sum of the product of the zeros taken two at a time) - ( product of the zeros)=0
=> x³ - (α+β+γ)x² + (αβ+βγ+γα)x - (αβγ)=0
=> x³ -2x²-7x+14=0 which is the required polynomial.
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:
(i) (ii)
(i) On comparing the given polynomial with the polynomial ax3 + bx2 + cx + d, we obtain a = 2, b = 1, c = -5, d = 2
Also,
2+1-5+2 = 0
Also,
-16+4+10+2=0
Hence are the zeroes.
α= β = 1 γ= -2
α+β+γ = =
αβ+βγ+γα =
αβγ =
Thus, the relationship between the zeroes and the coefficients is verified.
(ii) On comparing the given polynomial with the polynomial ax3 + bx2 + cx + d, we obtain a = 1, b = -4, c = 5, d = -2.
Hence 2, 1 and 1 are the zeroes.
α= 2 β = 1 γ= 1
α+β+γ =
αβ+βγ+γα =
αβγ =
Thus, the relationship between the zeroes and the coefficients is verified.
If the zeroes of the polynomial are a – b, a, a + b, find a and b.
Let p(x) = x3 - 3x2 + x + 1
The zeroes of the polynomial p(x) are given as a - b, a, a + b.
Comparing the given polynomial with dx3 + ex2 + fx + g, we can observe that
d = 1, e = -3, f = 1, g = 1
Sum of the zeroes = α+β+γ =
Product of the zeroes αβγ =
Substituting the value of a from (i) and (ii), we get,
Hence a=1 and
If two zeroes of the polynomial are
find other zeroes.
The 2 zeroes which are given are
Therefore, x2 - 4x + 1 is a factor of the given polynomial.
If the polynomial is divided by another polynomial
, the remainder comes out to be x + a, find k and a.
By dividing - 6x³ + 16 x² - 25x + 10 by x² - 2x + k we get remainder
( -9+2k )x+10 - 8k +
But ATQ, Remainder = x + a.
On comparing their coefficients,we have
2k - 9 = 1 ⇒ 2k = 10 ∴ k = 5
and, - ( 8 - k )k + 10 = a
⇒ a = - ( 8 - 5 )5 + 10
= - 3 × 5 + 10 = - 15 + 10 = - 5
∴ k = 5 & a = - 5
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