الأسبوع 3 · مجموعة الأصفار الحقيقية لدوال كثيرات الحدود: درجة الدالة والأصفار جبريًا وبيانيًا وإيجاد معامل مجهول · 3 دروس
2
Finding the set of zeros by solving f(x) = 0 with factorising
Mathematics Prep 3 Term 1, Unit 2, Lesson 2
GoalFind the set of real zeros of a polynomial function by factorising, including the cases with no zero.
New words
set of zeros · all real numbers x with f(x) = 0, written in bracesempty set ∅ · the set with no members, used when there is no real zerocommon factor · a factor shared by all the terms, such as x in x³ − 9xzero-product rule · if A × B = 0 then A = 0 or B = 0
Explanation
To find the zeros of f, solve f(x) = 0. Take out any common factor, factorise what is left, and then set each factor equal to zero (if A × B = 0, then A = 0 or B = 0). Never divide the equation by x: that would lose the zero x = 0. Some polynomials have no real zero. Because a square is never negative, x² + 4 is always at least 4, so 5x² + 20 = 0 has no solution and its set of zeros is ∅ (not {0}). Also x³ − 8 = 0 has only one real zero, x = 2, because x³ − 8 = (x − 2)(x² + 2x + 4) and x² + 2x + 4 = (x + 1)² + 3 is never zero. Write the answer as a set, for example {−5, 2}.
f(x)
Factorised
Set of zeros
2x − 10
2(x − 5)
{5}
x² − 49
(x − 7)(x + 7)
{−7, 7}
x² + 3x − 10
(x + 5)(x − 2)
{−5, 2}
x³ − 9x
x(x − 3)(x + 3)
{−3, 0, 3}
5x² + 20
5(x² + 4)
∅
Worked examples
Find the zeros of f(x) = x² + 3x − 10. Solve x² + 3x − 10 = 0. Find two numbers with product −10 and sum 3: 5 and −2. So (x + 5)(x − 2) = 0, giving x = −5 or x = 2. The set of zeros is {−5, 2}. Check: f(2) = 4 + 6 − 10 = 0 and f(−5) = 25 − 15 − 10 = 0.
Find the zeros of f(x) = x³ − 9x. Take out x: x(x² − 9) = x(x − 3)(x + 3) = 0. So x = 0, x = 3 or x = −3. The set is {−3, 0, 3}. If we had divided by x we would have found only ±3 and lost 0.
Find the zeros of 5x² + 20 and of x³ − 8. 5x² + 20 = 5(x² + 4) and x² + 4 is at least 4, so there is no real zero: ∅. x³ − 8 = (x − 2)(x² + 2x + 4); the second factor is never 0, so the only zero is 2: {2}.
Class activity
Mistake hunt. A friend solves x³ − 9x = 0 by dividing both sides by x and gets {−3, 3}. Test x = 0 in the original equation. What went wrong, and what is the correct set?
