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?

Answer:

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 \(\left[0,4\right]\).

notion of polynomial.png

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