Question

POSSIBLE POINTS: 13Part A: Which polynomial below is a fourth-degree polynomial in standard form? Explain how you know it is a fourth-degree polynomial and how doyou know it's in standard form. (7 points)a. 2x³ +-3x² +1 +5x*b. 5x + 2x³-3x² + 1C.-3x² +1+5x*+ 2x³d. 1-3x²+2x³ + 5x*Part B: Explain the closure property as it relates to polynomials and give an example. (6 points)

Answer 1

Given

The polynomials,

a. 2x³ +-3x² +1 +5x⁴

b. 5x⁴ + 2x³-3x² + 1

c.-3x² +1+5x⁴+ 2x³

d. 1-3x²+2x³ + 5x⁴

To find:

A) Which polynomial is a fourth-degree polynomial in standard form? Explain how you know it is a fourth-degree polynomial and how do you know it's in standard form.

B) Explain the closure property as it relates to polynomials and give an example.

Step-by-step explanation:

It is given that,

The polynomials,

a. 2x³ +-3x² +1 +5x⁴

b. 5x⁴ + 2x³-3x² + 1

c.-3x² +1+5x⁴+ 2x³

d. 1-3x²+2x³ + 5x⁴

Part A)

From, the above polynomials,

b. 5x⁴ + 2x³-3x² + 1 is the fourth degree polynomial in its standard form.

Reason:

Because the standard form of a polynomial starts with its highest degree and ends with lowest degree in a descending order.

Part B)

Also,

Closure property of a polynomial states that,

The addition or subtraction of two polynomials is again a polynomial.

For example,

Consider two polynomials,

p(x) = 5x⁴ + 2x³-3x² + 1, and q(x) = 5x⁴ + 3x³-2x² + 1

That implies,

`\begin{gathered} p(x)+q(x)=(5x⁴+2x³-3x²+1)+(5x⁴+3x³-2x²+1) \\ =(5+5)x⁴+(2+3)x³+(-3-2)x²+(1+1) \\ =10x⁴+5x³-5x²+2 \\ which\text{ }is\text{ }also\text{ }a\text{ }polynomial. \end{gathered}`