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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`
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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.
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