Calculate. y = 3 x 5 + 5 x 4 − 2 x 3

Question

Calculate.   y  =  3      x          5        +  5      x          4        −  2      x          3

Brenton Gay · Accepted Answer

X intercepts coorespond to the x coordinates where y = 0. Setting the function equal to 0

3 x^{5} + 5 x^{4} - 2 x^{3} = 0

Factoring out x cubed

x^{3} (3 x^{2} + 5 x - 2) = 0

Factoring our quadratic

x^{3} (3 x - 1) (x + 2) = 0

the whole thing will be zero if any of the above factors are zero

x^{3} = 0

x=0
3x-1=0
3x=1

x = \frac{1}{3}

x+2=0
x=-2
So the x intercepts are -2, 0, 1/3

Bruno Thompson · Accepted Answer

Substitute   z  =      x          4      . Then, the equation reads   y  =  3      z          2        +  5  z  −  2.Using the Pythagorean theorem (  z  =  [  −  b  ±            b              2              −    4    a    c    ]      /    2  a, where a = 3, b = 5, and c = -2:   z  =  [  −  5  ±      52    −    (    4    ∗    3    ∗    −    2    )    ]      /    (  2  ∗  3  )  =  [  −  5  ±      25    +    24    ]      /    6  =  [  −  5  ±      49    ]      /    6  =  [  −  5  ±  7  ]      /    6 produces two solutions: 2/6 = 1/3 OR -12/6 = -2.Therefore, the two z-intercepts of the substituted equation are z =1/3 and z = -2.However, since we substituted   z  =      x          4       earlier, now to find our x-intercepts, we have   x  =  (  1      /    3      )          4           a  n  d     x  =  (  −  2      )          4      . Thus, our solutions are x= 1/81 and x = 16.

Calculate. y=3x^5 +5x^4 -2x^3

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