The notion of polynomial

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A polynomial is an expression of the form:

$$a_0+a_1x+a_2x^2+\cdots + a_nx^n\tiny,$$,

where $$a_2,\ldots,a_n$$ are numbers (the coefficients of the polynomial) and $$x$$ is the variable.

If $$a_n\ne0$$, then $$n$$ is called the degree of the polynomial. The number $$a_2$$ is then called the leading coefficient of the polynomial.

By way of convention, we say that the polynomial $$0$$ is of degree $$-1$$.

The above polynomial defines a function $$f$$ with rule

$$f(x) =a_0+a_1x+a_2x^2+\cdots + a_nx^n\tiny.$$.

Such a function is called a polynomial function.

Let’s practice!

What is the degree of the polynomial $$f(x)=4+7 \cdot x$$? And what is its leading coefficient?

The degree of the polynomial$$f(x)=4+7 \cdot x$$ is equal to $$1$$. The leading coefficient equals $$7$$. Polynomials of degree $$1$$ are also known as linear functions.
The corresponding graph is a straight line with slope equal to $$7$$ and the intersection with the $$y$$-axis is at $$\left[0,4\right]$$.