Could someone please help me with this?
Is it possible to subtract two polynomials, each of degree 3, and have the difference be a polynomial of degree 2? If so, give an example. If not, explain why not.
thanks
(x^3+ 5x^2+4) - (x^3 - 3x^2 - 6x)
= x^3 + 5x^2 + 4 - x^3 + 3x^2 + 6x
= 8x^2 + 6x + 4
mmmhhh?
Certainly! Let's go through the process step by step to understand whether it is possible to subtract two degree 3 polynomials and obtain a polynomial of degree 2.
To subtract two polynomials, we subtract the corresponding terms of each polynomial. Since we have polynomials of degree 3, let's label them as follows:
First polynomial: Ax^3 + Bx^2 + Cx + D
Second polynomial: Ex^3 + Fx^2 + Gx + H
To find the difference of the two polynomials, we subtract the corresponding coefficients of each term. The result will be a new polynomial with the same degree as the original polynomials. Let's write the difference:
(Ax^3 + Bx^2 + Cx + D) - (Ex^3 + Fx^2 + Gx + H)
Now, let's simplify the expression:
= (Ax^3 - Ex^3) + (Bx^2 - Fx^2) + (Cx - Gx) + (D - H)
To obtain a polynomial of degree 2, we need the term with the highest degree, i.e., x^3, to cancel out. This can only happen if the coefficients of Ax^3 and Ex^3 are equal. In other words, A must be equal to E.
However, if A and E are equal, the resulting polynomial will have a degree of 3 and not 2. This is because, even after canceling out the x^3 term, we will still have x^2, x, and constant terms.
Therefore, it is not possible to subtract two polynomials, each of degree 3, and obtain a polynomial of degree 2. The result will always have a degree of 3 or higher due to the nature of polynomial subtraction.