What is the the degree of polynomial P defined by : P(x) = -5(x - 2)(x^3 + 5) + x^5?

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The degree of a polynomial is the largest exponent when the polynomial is written in standard form. So, expand this polynomial into standard form:

5(x - 2)(x3 + 5) + x5 =

x^5+5 x^4-10 x^3+25 x-50

So this polynomial is of degree 5.

The degree of a polynomial is the highest power of x.

For example:

f(x) = x^3 -4x^2 +1 ==> f(x) is a third degree polynomial.

To determine the degree of the given polynomila, we will need to open the brackets and rewrite into terms.

Let us open the brackets.

==> P(x) = -5(x-2)(x^3 + 5) + x^5

==> P(x) = -5 (x^4 + 5x^2 -2x^3 -10) + x^5

==> P(x) = x^5 - 5x^4 -+10x^3 -25x^2 -10

We notice that the highest power is x^5

**Then the polynomial is a fifth degree.**

We have P(x) = -5(x - 2)(x^3 + 5) + x^5

P(x) = -5(x - 2)(x^3 + 5) + x^5

=> P(x) = (-5x + 10)(x^3 + 5) + x^5

=> P(x) = -5x^4 - 25x + 10x^3 + 50 + x^5

=> P(x) = x^5 - 5x^4 + 10x^3 - 25x + 50

The degree of a polynomial is the highest power of x in the expression.

Here the degree is **5**

To find the degree of the polynomial P(x) = -5(x - 2)(x^3 + 5) + x^5.

The degree of the polynomial is the degree of the highest term.

So we expand the right side:

P(x) = -5(x-2)(x^3+5) +x^5

P(x) = -5(x^4 +5x-2x^3-10)+x^5.

P(x) = -5(x^4-2x^3+5x-10) +x^5.

P(x) = -5x^4+10x^3-25x+50+x^5.

We arrange the terms on the right side.

P(x) = x^5 -5x^4 +10x^3-25x+50

The highest term is x^5 with degree 5.

**So the degree of the polynomial is 5.**

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