Recognise and classify a polynomial

Polynomial classification

A polynomial in a named variable has only non-negative integer powers of that variable. Once the terms are ordered by power, its degree, leading coefficient and constant coefficient can be read directly

1. Check variable exponents Every exponent of the named variable must be a non-negative integer \(\displaystyle x^0,\quad x^2,\quad x^5\text{ are allowed powers}\)
2. Allow real coefficients Coefficients may be integers, fractions or other real numbers \(\displaystyle \frac{1}{2}x^2+\sqrt{2}\text{ is a polynomial in }x\)
3. Read the greatest exponent After arranging the terms if needed, the greatest exponent is the degree \(\displaystyle 3x^4-2x+1\Rightarrow\text{degree}=4\)
4. Use zero for missing powers An absent power has coefficient zero, including an absent constant term \(\displaystyle 5x^3-2x=5x^3+0x^2-2x+0\)

Worked example

Determine whether the expression below is a polynomial in t, then state its degree and coefficients

1. \(\displaystyle t^1,\quad t^0,\quad t^4\text{ all have non-negative integer exponents}\)Every exponent of the named variable satisfies the polynomial condition
2. \(\displaystyle R(t)=\frac{1}{2}t^4+3t-4\)Arrange the terms in descending order of power before reading the features
3. \(\displaystyle \text{degree}=4\)The greatest exponent is 4.
4. \(\displaystyle \text{leading coefficient}=\frac{1}{2},\qquad\text{coefficients of }t^3,t^2,t\text{ are }0,0,3,\qquad\text{constant coefficient}=-4\)The leading coefficient belongs to t⁴, and the constant coefficient is the term without t.

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