Mathematics

Problem

Consider the polynomial

P(x)=4+7(x-5)+2(x-5)^3.

Which statement correctly identifies the degree of

P

and justifies that choice? a)

P

is fourth degree because the exponents of

(x-5)^3

and

(x-5)

add to give

4

.},{ b)

P

is third degree because the highest non-zero power of

(x-5)

present is

3

. c)

P

is second degree since the

x^3

term produced by

(x-5)^3

cancels out when expanded and combined. d) The degree of

P

cannot be determined until all terms are expanded and simplified.

Intuitive explanation

The degree of a polynomial is determined by the highest power of the variable that appears when you expand it fully. Here, the $(x-5)^3$ term will expand to include an $x^3$ component, making that the dominant term—so the degree is 3. The key insight is that you don't need to fully expand everything; just identify which term, once expanded, contributes the highest power of $x$ .

Intuitive explanation

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