Question

show with a suitable example, that the sum of two monic polynomials need not to be a monic.​

Answer 1

Answer:

What is a univariate polynomial?

A univariate polynomial is a polynomial that only has one variable.

These are all univariate polynomials:

x³ + 3x² + 7x - 15 → x is the only variable

2y² - 14y + 13 → y is the only variable

a³ + a² + a + 1 → a is the only variable

This is NOT a univariate polynomial:

x² - y² → it has two variables, x and y

What is a monic polynomial?

A monic polynomial

has only one variable (a univariate polynomial)

the nonzero coefficient of the highest degree term is equal to 1.

These are both monic polynomials

x² + 7x + 14 → variable x → leading coefficient 1

y² + 8y- 28 → variable y → leading coefficient 1

These are NOT monic polynomials:

x² + y² → two variables

2x³ + x² + x + 1 → leading coefficient is 2

Must the sum of two monic polynomials always be monic?

Let’s add the two monic polynomials from above:

x² + 7x + 14

y² + 8y – 28

——————————

x² + y² + 7x + 8y – 14

This is NOT monic because it has two variables

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