Roots of polynomials questions

Questions

• 1. 3x^2+7x-5=0 as roots α and β. Find a quadratic with roots 1/α + 1/β
• 2. 4x^2+7x-5=0 has the roots α and β. Find a quadratic with the roots α^2 + β^2
• 3. 2x^2+7x-3=0 as the roots α and β. Find a quadratic with the roots α+5/β and β+5/α.
• 4. x^2-6x-3=0 has the roots α and β . By substitution, find an equation with the roots 2α-1 and 2β-1.
• 5. x^3-3x^2+2=0 has the roots α ,β and γ. By substitution or otherwise, find a cubic with the roots 3α , 3β and 3γ.
• 6. x^4+ax^3+14x^2-18x+b=0 has the roots 3i, -3i, 1-3i and 1+3i. Find a and b.

1....

• 1. 3x^2+7x-5=0 as roots α and β. Find a quadratic with roots 1/α + 1/β
• Σα = -b/a = -7/3 , product = -5/3
• New sum = 1/α + 1/β , α+β/αβ , -7/3 / -5/3 = 7/5
• New product = 1/α x 1/β , 1/αβ , 1/-5/3 = -3/5
• Create equation: x^2+7/5x-3/5 , or 5x^2+7-3=0

2....

• 2. 4x^2+7x-5=0 has the roots α and β. Find a quadratic with the roots α^2 + β^2
• Σα = -b/a = -7/4 , product = -5/4
• New sum: α^2 + β^2 , (α+β)^2-2αβ , (-7/4)^2 - 2(-5/4) =89/16
• New product: α^2 + β^2 , (αβ)^2 , (-5/4)^2 = 25/16
• Create equation: x^2+89/16x+25/16=0 , or 16x^2+89+25

3

• 3. 2x^2+7x-3=0 as the roots α and β. Find a quadratic with the roots α+5/β and β+5/α.
• Σα = -b/a = -7/2 , product = -3/2
• New sum = α+5/β + β+5/α , 5(α+β)/αβ + (α+β) , 5(-7/2)/-3/2 + (-7/2) = 49/6
• New product = α+5/β x β+5/α , (α+5/β)(β+5/α) , αβ+10+25/αβ , -3/2+10+25/-3/2 = -49/6.
• Create equation: x^2+49/6-49/6, or 6x^2+49x-49

4....

• 4. x^2-6x-3=0 has the roots α and β . By substitution, find an equation with the roots 2α-1 and 2β-1.
• We can say y=2x-1 , so therefore x=(y+1)/2
• Now we sub this into the original equation, so ((y+1)/2)^2 - 6((y+1)/2) -3 = 0 and this can be simplified to get the equation y^2-10y-23=0
• Remember to rewrite in terms of x, so x^2-10x-23=0

5....

• 5. x^3-3x^2+2=0 has the roots α ,β and γ. By substitution or otherwise, find a cubic with the roots 3α , 3β and 3γ.
• We can say y=3x, so x=y/3.
• Now we sub this into the original equation, so (y/3)^3 -3(y/3)^2 +2=0 and this can be simplified to give y^3-9y^2+54
• Remember to rewrite in terms of x, so x^3-9x^2+54

6....

• 6. x^4+ax^3+14x^2-18x+b=0 has the roots 3i, -3i, 1-3i and 1+3i. Find a and b.
• Σα = -b/a = -a , product = e/a = b
• Now calculate the sum of the roots: (3i)+(-3i)+(1+3i)+(1-3i) = 2, so -a=2, so a=-2
• Now calculate the product of the roots: (3i)(-3i)(1+3i)(1-3i) , (-9i^2)(-9i^2+1) , 9x10 = 90, b=90