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Answer :
The given equation is [tex]20x^4 + 3x^3 + 31x - 38[/tex] .
The expression that must be subtracted from [tex]20x^4 + 3x^3 + 31x - 38[/tex] to get [tex]9x^4 + 23x^3 - 8x - 61[/tex] is [tex]11x^4 - 20x^3 + 39x - 99[/tex].
To find what must be subtracted from the expression [tex]20x^4 + 3x^3 + 31x - 38[/tex] to get [tex]9x^4 + 23x^3 - 8x - 61[/tex], we need to set up an equation. Let's call the unknown value we need to subtract as "y".
We have the equation:
[tex](20x^4 + 3x^3 + 31x - 38) - y = 9x^4 + 23x^3 - 8x - 61[/tex]
To solve for y, we can combine like terms on both sides of the equation:
[tex](20x^4 - 9x^4) + (3x^3 - 23x^3) + (31x + 8x) - (38 + 61) = y[/tex]
Simplifying each term, we get:
[tex]11x^4 - 20x^3 + 39x - 99 = y[/tex]
So, the expression that must be subtracted from [tex]20x^4 + 3x^3 + 31x - 38[/tex] to get [tex]9x^4 + 23x^3 - 8x - 61[/tex] is [tex]11x^4 - 20x^3 + 39x - 99[/tex].
We begin by setting up an equation where y represents the unknown value to be subtracted. By subtracting y from the expression on the left side of the equation, we should obtain the expression on the right side of the equation.
Next, we combine like terms on both sides of the equation. We subtract the coefficients of the same power of x, and add the coefficients of the constant terms. This simplification helps us isolate the unknown value, y.
Finally, we simplify the equation further to obtain the expression that needs to be subtracted. In this case, the expression that must be subtracted is [tex]11x^4 - 20x^3 + 39x - 99[/tex].
Conclusion:
To obtain the expression [tex]9x^4 + 23x^3 - 8x - 61[/tex] from [tex]20x^4 + 3x^3 + 31x - 38[/tex], we need to subtract [tex]11x^4 - 20x^3 + 39x - 99[/tex].
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