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8x^2 + 6x - 1  =  0

ab  = -6/8 = -1/8

a + b  = -6/8 = -3/4          square both sides

a^2 + 2ab + b^2  = 9/16

a^2 + b^2   = 9/16  - 2ab

a^2 + b^2  = 9/16  +1/4

a^2 + b^2  =  13/16

a^3 + b^3  = 

(a + b) ( a^2 + b^2  -  ab)  =

(-3/4) (13/16 + 1/8)  = (-3/4)( 13/16 + 2/16)   (-3/4) ( 15/16)  = -45 / 64

Absolutely, I’ve been improving my problem-solving abilities in solving polynomial equations. Let's find a3+b3, where a and b are the roots of the equation: 5x2−11x+4=−3x2−17x+5

We can solve the equation for a and b using the quadratic formula and then use the fact that for any quadratic equation ax2+bx+c=0, the sum of the roots is −ab​ and the product of the roots is ac​.

Steps to solve: 1. Solve the equation for a and b: 5x2−11x+4=−3x2−17x+5

Combining like terms and rearranging the equation, we get: 8x2+6x−1=0

Using the quadratic formula, we get: x=2a−b±b2−4ac​​ where a, b, and c are the coefficients of the quadratic equation. In this case, a = 8, b = 6, and c = -1. Substituting these values into the formula, we get:

x=2⋅8−6±62−4⋅8⋅−1​​

x=16−6±217​​

Therefore, the roots are x=817​−3​ and x=8−17​−3​.

2. Find a^3 + b^3: We know that for any quadratic equation ax2+bx+c=0, the sum of the roots is −ab​ and the product of the roots is ac​. In this case, we have:

a+b=−86​=−43​ ab=8−1​

We can use these relationships to find a3+b3:

a3+b3=(a+b)(a2−ab+b2)

Substitute the values we found for a + b and ab:

a3+b3=−43​(a2−ab+b2)

We don't need to find the individual values of a^2 and b^2$ since we can rewrite the expression using the fact (a+b)2=a2+2ab+b2 :

(a+b)2=a2+2ab+b2

Substitute a + b = -3/4:

(−43​)2=a2+2ab+b2

Expand:

169​=a2+2ab+b2

Substitute ab = -1/8:

169​=a2−41​+b2

Combine like terms:

169​+41​=a2+b2

a2+b2=45​

Substitute this back into the expression for a^3 + b^3:

a3+b3=−43​(45​−81​)

a3+b3=−43​⋅89​

a3+b3=−3227​

Answer: a3+b3=−3227​