To find the difference between the two given polynomials, we need to subtract the second polynomial from the first. Let's write down each polynomial clearly:
1. First polynomial: \( x^3 + 4x^2 - 7x - 2 \)
2. Second polynomial: \( 2x^2 - 9x + 4 \)
Now, perform the subtraction term-by-term:
1. Cubic term:
[tex]\[
x^3 - 0 = x^3
\][/tex]
(Note that the second polynomial has no \( x^3 \) term, so it is effectively \( 0 \) for the cubic term.)
2. Quadratic term:
[tex]\[
4x^2 - 2x^2 = 2x^2
\][/tex]
3. Linear term:
[tex]\[
-7x - (-9x) = -7x + 9x = 2x
\][/tex]
4. Constant term:
[tex]\[
-2 - 4 = -6
\][/tex]
Putting it all together, the resulting polynomial after subtraction is:
[tex]\[
x^3 + 2x^2 + 2x - 6
\][/tex]
So, the coefficients for each term from highest degree to the constant term are:
[tex]\[
[1, 2, 2, -6]
\][/tex]
Thus, the difference between the given polynomials is:
[tex]\[
x^3 + 2x^2 + 2x - 6
\][/tex]
