Answer:
1. Fill in the box with 1
2. Fill in the box with -2
Step-by-step explanation:
Expression:
[tex](-2x^3 + [\ ]x)(x^{[\ ]}+1.5) = A[/tex]
Solving (1): Fill in the box to make it a polynomial.
To make it a polynomial, we simply fill in the box with a positive integer (say 1)
Fill in the box with 1
[tex](-2x^3 + [1]x)(x^{[1]}+1.5) = A[/tex]
Remove the square brackets
[tex](-2x^3 + x)(x^1+1.5) = A[/tex]
[tex](-2x^3 + x)(x+1.5) = A[/tex]
Open bracket
[tex]-2x^4 - 3x^3 + x^2 + 1.5x = A[/tex]
Reorder
[tex]A = -2x^4 - 3x^3 + x^2 + 1.5x[/tex]
The above expression is a polynomial.
This will work for any positive integer filled in the box
Solving (2): Fill in the box to make it not a polynomial.
The powers of a polynomial are greater than or equal to 0.
So, when the boxes are filled with a negative integer (say -2), the expression will cease to be a polynomial
Fill in the box with -2
[tex](-2x^3 + [-2]x)(x^{[-2]}+1.5) = A[/tex]
Remove the square brackets
[tex](-2x^3 - 2x)(x^{-2}+1.5) = A[/tex]
Reorder
[tex]A = (-2x^3 - 2x)(x^{-2}+1.5)[/tex]
Open brackets
[tex]A = -2x-3x^3-2x^{-1}-3x[/tex]
Collect Like Terms
[tex]A = -3x^3-2x-3x-2x^{-1}[/tex]
[tex]A = -3x^3-5x-2x^{-1}[/tex]
Notice that the least power of x is -1.
Hence, this is not a polynomial.
