Question

Here are two expressions whose product is a new expression, :

Answer 1

Answer:

1. Fill in the box with 1

2. Fill in the box with -2

Step-by-step explanation:

Expression:

$(-2x^3 + [\ ]x)(x^{[\ ]}+1.5) = A$

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

$(-2x^3 + [1]x)(x^{[1]}+1.5) = A$

Remove the square brackets

$(-2x^3 + x)(x^1+1.5) = A$

$(-2x^3 + x)(x+1.5) = A$

Open bracket

$-2x^4 - 3x^3 + x^2 + 1.5x = A$

Reorder

$A = -2x^4 - 3x^3 + x^2 + 1.5x$

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

$(-2x^3 + [-2]x)(x^{[-2]}+1.5) = A$

Remove the square brackets

$(-2x^3 - 2x)(x^{-2}+1.5) = A$

Reorder

$A = (-2x^3 - 2x)(x^{-2}+1.5)$

Open brackets

$A = -2x-3x^3-2x^{-1}-3x$

Collect Like Terms

$A = -3x^3-2x-3x-2x^{-1}$

$A = -3x^3-5x-2x^{-1}$

Notice that the least power of x is -1.

Hence, this is not a polynomial.