If α and β are the zeroes of the polynomial x2
– 7x + 10, then find the value of α3 + β3
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Answer:
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If alpha and beta are the zeros of the quadratic polynomial x^2 +7x+10, find the value of alpha^3 +beta^3?
[math]x^2 + 7x + 10 = 0[/math]
[math](x + 2) (x + 5) = 0[/math]
[math]\alpha, \beta = -2 \; or \; -5[/math]
[math]\alpha^3 + \beta^3 = (-2)^3 + (-5)^3 = (-8) + (-125) = \boxed{\boldsymbol{-133}}[/math]
The following alternate method can be used without actually calculating zeros. This method is especially useful when we have more complicated zeros.
For quadratic polynomial: [math]\; ax^2 + bx + c[/math]
[math]\quad\quad\quad [/math]Sum of zeros [math]= -\frac{b}{a}[/math]
[math]\quad\quad\quad [/math]Product of zeros [math]= \frac{c}{a}[/math]
Quadratic polynomial [math]\; x^2 + 7x + 10 \;[/math] has zeros [math]\;\alpha\;[/math] and [math]\;\beta[/math]
[math]\quad\quad\quad \alpha + \beta = -7[/math]
[math]\quad\quad\quad \alpha \beta = 10[/math]
[math](\alpha + \beta)^3 = (-7)^3[/math]
[math]\alpha^3 + 3\alpha^2\beta + 3\alpha\beta^2 + \beta^3 = -343[/math]
[math]\alpha^3 + 3\alpha\beta(\alpha+\beta) + \beta^3 = -343[/math]
[math]\alpha^3 + 3(10)(-7) + \beta^3 = -343[/math]
[math]\alpha^3 - 210 + \beta^3 = -343[/math]
[math]\alpha^3 + \beta^3 = -343 + 210[/math]
[math]\boxed{\boldsymbol{\alpha^3 + \beta^3 = -133}}[/math]
Step-by-step explanation:Hii aapko kitni lang. aati hai