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Polynomial Basics

What are Polynomials?

Polynomials are functions of the form: $$\square x^n+\square x^{n-1}+\square x^{n-2}+\cdots+\square x^2+\square x+\square$$ where the powers of $x$ are positive integers.
Consider the polynomial, $$\displaystyle P(x)=8x^6+\frac{1}{2}x^5-7x^4+\sqrt{3}x^2-x+1$$

The degree of $P(x)$ is

The coefficient of the $x^5$ term is

The coefficient of the $x^4$ term is

The coefficient of the term is $\sqrt{3}$

The coefficient of the $x$ term is

The leading coefficient of $P(x)$ is

The constant term of $P(x)$ is

Which of the following functions are polynomials?

$P(x)=-4x^4-2x^2+x$ 

$Q(x)=1+x-2x^2-4x^4$ 

$R(x)=-4x^4-2x^2+x+1+x^{-1}$ 

$f(x)=-4x^4+1$ 

$g(x)=-4x^4-2x^\frac{1}{2}$ 

$h(x)=x+1$ 

$j(x)=3^x+x^2-x+1$ 

 

Operations with Polynomials

If $P(x)=4x^4+3x^3-2x^2-x-1$ and
$\;\; Q(x)=5x^4-2x^3-2x^2+x+3$, then

$P(x)+Q(x)=$ $x^4+x^3-$ $x^2+$

$P(x)-Q(x)=$ $x^4+$ $x^3-$ $x-$

$2P(x)-3Q(x)=$ $x^4+$ $x^3+$ $x^2-$ $x-$

If $P(x)=2x^3-4x^2+3$ and $Q(x)=x^4-3x^3+2x-6$, then

$P(x)+Q(x)=$

$P(x)-Q(x)=$

$3P(x)+2Q(x)=$