What happens if we expand, then differentiate?

What happens if we expand, then differentiate?

Question

The expression $\frac{d^2}{dx^2}\bigl[(2x-1)^4(1-x)^5\bigr] + \frac{d}{dx}\bigl[(2x+1)^4(3x^2-2)^2\bigr]$ is a polynomial of degree

1. $9$;

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1. $8$;

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1. $7$;

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1. less than $7$.

Solution

Let’s observe that

1. multiplying two polynomials of degrees $p$ and $q$ gives a polynomial of degree $p+q$, and

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1. differentiating a polynomial of degree $p$ gives a polynomial of degree $p-1$.

From this we can quickly see that the expression in the question is a sum of two polynomials, each of degree 7.

Does this mean their sum has to be a polynomial of degree 7 too?

Yes, unless the leading terms of the two expressions cancel when you add them together.

So what are our leading terms here?

The expression $(2x-1)^4(1-x)^5$ gives first term $(2x)^4(-x)^5=-16x^9$, which differentiates twice to $-9 \times 8 \times 16 x^7.$

The leading term of the expression $(2x+1)^4(3x^2-2)^2$ is $(2x)^4(3x^2)^2=16 \times 9x^8$, which differentiates to $16 \times 9 \times 8x^7.$

So in fact the two leading terms here DO cancel, and we’re left with a polynomial of degree less than $7$, so the answer is (d).