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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
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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"`
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