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.
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To determine whether it is possible to subtract two polynomials of degree 3 and obtain a polynomial of degree 2, we need to understand the rules of polynomial subtraction and the degree of resulting polynomials.
When subtracting two polynomials, we subtract the corresponding terms. For example, if we subtract the polynomial p(x) = ax^3 + bx^2 + cx + d from the polynomial q(x) = px^3 + qx^2 + rx + s, the resulting polynomial will be (q - p)(x) = (px^3 + qx^2 + rx + s) - (ax^3 + bx^2 + cx + d).
To find the degree of the resulting polynomial, we need to identify the highest degree term after subtracting. In this case, the highest degree term will occur when we subtract the leading terms of each polynomial, i.e., the terms with the highest exponents.
So, let's subtract the leading terms: px^3 - ax^3 = (p - a)x^3.
From this, we can see that if the subtracted leading term results in a non-zero coefficient with x^3, then the resulting polynomial will also have a degree of 3. This is because when we subtract the leading terms, we either get a non-zero coefficient for the highest degree term, or we get zero.
Therefore, it is not possible to subtract two polynomials of degree 3 and yield a polynomial of degree 2. The resulting polynomial will either have a degree of 3 or higher.