الأسبوع 3 · مجموعة الأصفار الحقيقية لدوال كثيرات الحدود: درجة الدالة والأصفار جبريًا وبيانيًا وإيجاد معامل مجهول · 3 دروس
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Polynomial functions: degree, constant and zero polynomial, and what a zero of a function is
Mathematics Prep 3 Term 1, Unit 2, Lesson 2
GoalDecide whether a rule is a polynomial function and give its degree, and check whether a number is a zero.
New words
polynomial function · f(x) = a₀ + a₁x + … + aₙxⁿ, with whole-number exponents; its degree is ndegree · the highest exponent of x that has a non-zero coefficientconstant function · f(x) = 7: a polynomial of degree 0; the zero function f(x) = 0 has no degreezero of a function · a number a with f(a) = 0
Explanation
A polynomial function adds up terms of the form number × xⁿ, where each exponent n is 0, 1, 2, 3 … (a whole number). Its degree is the highest exponent: 3x − 6 is linear (degree 1), x² − x is quadratic (degree 2), 2x³ + x is cubic (degree 3), and a number such as 9 is a constant of degree 0. The zero function f(x) = 0 has no degree. Rules like 1 ÷ x, √x and 2ˣ are not polynomial functions, because x appears in a denominator, under a root or in an exponent. A zero of f is a number a such that f(a) = 0. Three statements then say the same thing: a is a zero of f; x = a solves f(x) = 0; and (x − a) is a factor of f(x). For example, x² − 7x + 10 = (x − 2)(x − 5), so its zeros are 2 and 5.
Rule
Polynomial?
Degree
3x − 6
yes
1
2x³ + x
yes
3
9
yes
0
1 ÷ x
no
–
√x
no
–
2ˣ
no
–
Worked examples
Decide which are polynomial functions and give the degree: 4x² − x, 7, x⁻² + 1, 5x⁴ + 3x. 4x² − x is a polynomial of degree 2. 7 is a polynomial of degree 0. x⁻² + 1 is not, because −2 is not a whole number ≥ 0 (it is 1 ÷ x² + 1). 5x⁴ + 3x is a polynomial of degree 4.
Which of 2, −2 and 1 are zeros of f(x) = x³ − 4x? f(2) = 8 − 8 = 0 ✓. f(−2) = −8 + 8 = 0 ✓. f(1) = 1 − 4 = −3, not zero. So 2 and −2 are zeros and 1 is not.
Compare g(x) = 5 and h(x) = 0. g(x) = 5 is never 0, so it has no zeros (its set of zeros is ∅). h(x) = 0 is zero for every x, so every real number is a zero (the set of zeros is R).
Class activity
Test and find. For f(x) = x³ − 6x² + 11x − 6 test x = 1, 2, 3 and 4. Which are zeros? Then check that f(x) = (x − 1)(x − 2)(x − 3) by expanding.
Practice
1
What is the degree of f(x) = 7x⁴ − x² + 3?
The highest exponent is 4, so the degree is 4.
2
Is f(x) = 3x² + √x a polynomial function? Why?
No. √x = x^(1/2), and 1/2 is not a whole number.
3
Is 3 a zero of f(x) = x² − 2x − 3?
f(3) = 9 − 6 − 3 = 0, so yes.
4
Is −1 a zero of f(x) = x³ + 2x² − x − 1?
f(−1) = −1 + 2 + 1 − 1 = 1, not 0. So no.
5
Give the set of zeros of f(x) = 9 and of f(x) = 0.
