Algebraic Number Theory - Papers Contributed for the Kyoto by S. Iyanaga (Editor)

By S. Iyanaga (Editor)

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2] (iii)Find the quadratic equation which has roots α + β and γ + δ. [3] (iv) Find α + β and γ + δ. (v) Show that α 2 − 3(1 + 10)α + 4 = 0 , and find similar quadratic equations satisfied by β, γ and δ.

Further Pure Mathematics 1 Algebra Chapter assessment 1. Find the values of the unknown constants in each of these identities. (i) 3 x 2 − 6 x − 1 ≡ A( x + B) 2 + C [3] (ii) 3 x − 2 ≡ P( x + 1) 2 + Q( x + 1)( x − 3) + R( x − 3) 2 [3] (iii) x 3 + 2 x − 3 ≡ ( x − 2)( Ax 2 + Bx + C ) + D [4] 2. The quadratic equation 2 z 2 − 4 z + 5 = 0 has roots α and β. (i) Write down the values of α + β and αβ. (ii) Find the quadratic equation with roots 3α – 1, 3β – 1. (iii)Find the cubic equation which has roots α, β and α + β.

I) 3 x 2 − 6 x − 1 ≡ A( x + B) 2 + C [3] (ii) 3 x − 2 ≡ P( x + 1) 2 + Q( x + 1)( x − 3) + R( x − 3) 2 [3] (iii) x 3 + 2 x − 3 ≡ ( x − 2)( Ax 2 + Bx + C ) + D [4] 2. The quadratic equation 2 z 2 − 4 z + 5 = 0 has roots α and β. (i) Write down the values of α + β and αβ. (ii) Find the quadratic equation with roots 3α – 1, 3β – 1. (iii)Find the cubic equation which has roots α, β and α + β. [2] [3] [4] 3. The equation z 3 + kz 2 − 4 z − 12 = 0 has roots α, β and γ. (i) Write down the values of αβ + βγ + γα and αβγ, and express k in terms of α, β and γ.

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