What this lesson unlocks
Polynomial language lets us describe expressions precisely. You will recognise polynomials, order their terms, evaluate them and decide whether a number is a zero.
Learning goals
- Identify whether an algebraic expression is a polynomial and explain the exponent rule.
- Write a polynomial in standard form and identify its degree and type.
- Evaluate p(k) and determine whether k is a zero.
Concept lesson
Terms, coefficients and constants
In 4x^3 - 2x + 9, the terms are 4x^3, -2x and 9; their powers are 3, 1 and 0. The constant 9 may be written 9x^0.
The polynomial test
A polynomial in x is a finite sum of terms ax^n where n is a non-negative integer. Thus 3x^2-5x+1 is a polynomial, while 1/x=x^-1 and square root of x=x^(1/2) are not.
Degree, type and standard form
Write terms in descending powers. The greatest exponent with a non-zero coefficient is the degree. Degree 1, 2 and 3 polynomials are linear, quadratic and cubic respectively. A non-zero constant has degree 0; the zero polynomial's degree is not defined.
Value and zero
p(k) is obtained by replacing every x with k. A number k is a zero exactly when p(k)=0. For ax+b, a not equal to zero, the zero is -b/a.
Order and classify a polynomial
Write 7-3x^3+2x in standard form and identify its degree and type.
Evaluate at a negative value
For p(x)=2x^2-3x+1, find p(-2).
Check a proposed zero
Verify whether 2 is a zero of p(x)=x^2-5x+6.
Polynomial or not?
Classify 5x^4-2, 3/x+x, square root of x+1, and 8. For each rejection, name the exponent that breaks the rule.
Substitute carefully
For p(x)=x^3-4x, calculate p(-2), p(0) and p(2). Decide which inputs are zeroes. Put each substituted negative input in brackets.
Repair a classmate's reasoning
A student says 6x^2+x^-1 has degree 2 because 2 is the largest exponent. Correct the statement by applying the polynomial test before discussing degree.
One-mark degree and value patterns
For degree, first ensure standard form and then read the greatest exponent with non-zero coefficient. For p(k), substitute into every term and show one clean line of working.
Exit ticket
(1) Decide whether 2x^3+x^-2 is a polynomial. (2) Put 5-2x+x^4 in standard form and name its degree. (3) For p(x)=x^2-1, find p(-1) and verify whether -1 is a zero.