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Let x be real number such that x3+4x=8 , then the value of x7+64x2 is
A.136
B.146
C.128
D.156

Answer
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Hint: There can be several ways to solve a polynomial equation. You can use any of them but first, you would have to think of establishing a link between the given equation and the equation that you have to find. Once you find that out, you can substitute the given values whenever needed and find out the correct answer.

Complete step-by-step answer:
This question can be solved by two methods,
First approach –
We are given that x3+4x=8
On squaring both sides, we get –
 (x3+4x)2=(8)2x6+16x2+8x4=64
To make the above equation of degree 7, we multiply x on both sides,
 x7+16x3+8x5=64x
Add 16x3 on both sides
 x7+32x3+8x5=64x+16x3
Now we take 8x2 common on the left-hand side and 16 common on the right-hand side.
 x7+8x2(x3+4x)=16(x3+4x)
We know the value of x3+4x , substituting the value in the above equation –
 x7+8x2(8)=16(8)x7+64x2=128
Thus, we have got the required answer.
Second approach –
We have to find x7+64x2
 x7 can be written as x(x3)2 , so the above equation becomes -
 x×(x3)2+64x2
We are given that,
 x3+4x=8x3=84x
Using this value in the equation that we have to find
 x(84x)2+64x2=x(64+16x264x)+64x2=64x+16x364x2+64x2=16x3+64x=16(x3+4x)=16(8)=128
That is, x7+64x2=128
So, the correct answer is “Option C”.

Note: An expression composed of variables, constants and exponents, that contain only the operations of subtraction, addition, multiplication and division but there cannot be a variable in the denominator. The highest exponent in the polynomial equation is called the degree of that equation. While taking commons, keep the given values in mind so that after taking common, they can be easily substituted.
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