1. Create an expression, containing at least two variables, that can be factored using the sum of two cubes identity.
1 answer:
Remember that the sum of tow cubes identity is: 
So, to create our expression<span>, containing at least two variables, that can be factored using the sum of two cubes, we just need to replace </span>
and 
with tow monomials with a different variable:
and 
Lets replace those values in our identity:
Now that we have our expression, lets factor it using the sum of two cubes identity:
To verify if the factored form of our expression (right hand side) is equivalent to the original form (left hand side), we are going to expand the right hand side:
Since both sides of the equation are equal, we can conduce that the factored form of our expression is equivalent to the original expression.
So, to create our expression<span>, containing at least two variables, that can be factored using the sum of two cubes, we just need to replace </span>
Lets replace those values in our identity:
Now that we have our expression, lets factor it using the sum of two cubes identity:
To verify if the factored form of our expression (right hand side) is equivalent to the original form (left hand side), we are going to expand the right hand side:
Since both sides of the equation are equal, we can conduce that the factored form of our expression is equivalent to the original expression.
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<h3>Answer:</h3>
(x, y) ≈ (1.49021612010, 1.22074408461)
<h3>Explanation:</h3>This is best solved graphically or by some other machine method. The approximate solution (x=1.49, y=1.221) can be iterated by any of several approaches to refine the values to the ones given above. The values above were obtained using Newton's method iteration.
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Setting the y-values equal and squaring both sides of the equation gives ...
... √x = x² -1
... x = (x² -1)² = x⁴ -2x² +1 . . . . . square both sides
... x⁴ -2x² -x +1 = 0 . . . . . polynomial equation in standard form.
By Descarte's rule of signs, we know there are two positive real roots to this equation. From the graph, we know the other two roots are complex. The second positive real root is extraneous, corresponding to the negative branch of the square root function.