Let b be a constant. What is the smallest possible degree of the polynomial f(x) + b \cdot g(x), where f(x) = 2x^5 - 6x^4 - 4x^3 + 12x^2 + 7x - 5 and g(x) = x^6 - 17x^5 + 2x^4 + 6x^3 + 11x^2 - 8x + 1.

LiIIiam0216 Jun 23, 2024

#1**+1 **

pleeeez use latttexxxxxx

\(f(x) + b \cdot g(x),\space f(x) = 2x^5 - 6x^4 - 4x^3 + 12x^2 + 7x - 5,\space g(x) = x^6 - 17x^5 + 2x^4 + 6x^3 + 11x^2 - 8x + 1.\)

im pretty sure its just 6 since no matter what you do (multiplication wise) to g(x) it will always contain the \(x^6\) term. im probrably overlooking some major thing.

shmewy Jun 23, 2024

#2**+1 **

Since b is a constant, it must not contain any powers. This means b has no overall effect on the degree.

Thus, let's take a look at f(x) and g(x).

The degree of \( f(x) + b \cdot g(x)\) is essentially the larger degree for f(x) and b(x).

f(x) has a degree of 5, because the leading term is \(2x^5\)

g(x) has a degree of 6, because the leading term is \(x^6\)

Thus, the degree of the polynomial must have a degree of 6.

So the smallest possible degree is 6.

Thanks! :)

NotThatSmart Jun 23, 2024