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Chapter 2. Polynomials

 
 
Grade 10  ·  Mathematics  ·  Chapter 2

Polynomials

From degrees to zeroes, graphs to coefficients — understand every concept with clear explanations, original diagrams, and step-by-step solved examples.

📐 Types of Polynomials 0️⃣ Zeroes & Graphs 📈 Parabolas α β Sum & Product 🧱 Build a Polynomial

📐
Section 1

What Is a Polynomial?

A polynomial is an algebraic expression built from a variable (like x) using only addition, subtraction, multiplication, and non-negative whole-number powers. Each separate chunk separated by + or − is called a term.

The degree of a polynomial is the highest power of the variable appearing in it. This single number tells you most of what you need to know about the polynomial’s behaviour.

The four main types for Grade 10:

Linear (deg 1)Quadratic (deg 2)Cubic (deg 3)General (deg n)
Four types of polynomials: linear degree 1 with straight line graph, quadratic degree 2 with parabola, cubic degree 3 with S-curve, and general degree n
💡 Is it a polynomial or not?
A polynomial must have whole-number, non-negative powers of the variable. Expressions like 1/x, √x, and x⁻² are NOT polynomials because they contain negative or fractional powers. Examples like 3x² + 2x + 5 ARE polynomials.

Is It a Polynomial? — Quick Reference

✅ YES — These are polynomials
7x + 3 → degree 1 (linear)
2x² − 5x + 1 → degree 2 (quadratic)
x³ − 4x → degree 3 (cubic)
5 → degree 0 (constant)
x⁴ + 2x² − 3 → degree 4
❌ NO — These are NOT polynomials
1/(x+2) → variable in denominator
√x + 3 → fractional power (x^½)
x² + x⁻¹ → negative power
2^x + 1 → variable is the exponent
|x| + 5 → absolute value not allowed
WORKED EXAMPLE A
Find the degree and identify the type of these polynomials
p(x) = 4x + 7: Highest power = 1 → Linear polynomial, degree 1
q(y) = 3y² − 2y + 9: Highest power = 2 → Quadratic polynomial, degree 2
r(t) = 2t³ − t + 5: Highest power = 3 → Cubic polynomial, degree 3
s(x) = x⁴ − 3x² + 1: Highest power = 4 → Polynomial of degree 4
f(x) = −8: No variable, constant → Constant polynomial, degree 0

🔢
Section 2

Value of a Polynomial at a Given Point

To find the value of a polynomial p(x) at a specific number, you simply substitute that number in place of x and calculate. We write p(k) to mean “the value of p(x) when x = k”.

If p(x) = ax² + bx + c, then p(k) = ak² + bk + c
Just replace every x with the value k and compute!
WORKED EXAMPLE B
Find the value of p(x) = 3x² − 2x + 5 at x = 2 and x = −1
At x = 2:
p(2) = 3(2)² − 2(2) + 5
p(2) = 3(4) − 4 + 5
p(2) = 12 − 4 + 5 = 13

At x = −1:
p(−1) = 3(−1)² − 2(−1) + 5
p(−1) = 3(1) + 2 + 5
p(−1) = 3 + 2 + 5 = 10

WORKED EXAMPLE C
For p(x) = 2x³ − x² + 4x − 3, verify that p(1) and find p(0) and p(−2)
p(1): 2(1)³ − (1)² + 4(1) − 3 = 2 − 1 + 4 − 3 = 2
p(0): 2(0)³ − (0)² + 4(0) − 3 = 0 − 0 + 0 − 3 = −3 (always the constant term!)
p(−2): 2(−8) − (4) + 4(−2) − 3 = −16 − 4 − 8 − 3 = −31

p(1) = 2
p(0) = −3
p(−2) = −31

0️⃣
Section 3

Zeroes of a Polynomial — What They Mean

A zero of a polynomial p(x) is any real number k that makes p(k) = 0. In other words, a zero is an input that produces an output of zero. Zeroes are also called roots of the polynomial.

🔑 Definition
k is a zero of p(x) if and only if p(k) = 0
Geometrically: the x-coordinate where the graph of y = p(x) crosses or touches the x-axis
📊 How Many Zeroes?
Degree 1 (linear): exactly 1 zero
Degree 2 (quadratic): at most 2 zeroes
Degree 3 (cubic): at most 3 zeroes
Degree n: at most n zeroes
WORKED EXAMPLE D
Find the zeroes of the linear polynomial p(x) = 5x − 15
Set p(x) = 0:
5x − 15 = 0
5x = 15
x = 15 ÷ 5 = 3

Verify: p(3) = 5(3) − 15 = 15 − 15 = 0 ✓
General rule: zero of ax + b = −b/a = −(−15)/5 = 15/5 = 3

Zero of p(x) = 3
WORKED EXAMPLE E
Check if x = 2 and x = −3 are zeroes of p(x) = x² + x − 6
Check x = 2:
p(2) = (2)² + (2) − 6 = 4 + 2 − 6 = 0 ✓ → Yes, x = 2 is a zero

Check x = −3:
p(−3) = (−3)² + (−3) − 6 = 9 − 3 − 6 = 0 ✓ → Yes, x = −3 is a zero

Factoring to confirm:
x² + x − 6 = (x − 2)(x + 3) → zeroes: x = 2 and x = −3

Both x = 2 and x = −3 are zeroes

📈
Section 4

Geometrical Meaning of Zeroes — Graphs

The most powerful way to understand zeroes is geometrically. When you plot the graph of y = p(x), the zeroes are precisely the x-coordinates of the points where the curve meets the x-axis — this is true for every type of polynomial.

For a linear polynomial (degree 1), the graph is a straight line that crosses the x-axis at exactly one point. For a quadratic polynomial (degree 2), the graph is a parabola — a smooth U-shape that opens upward (if a > 0) or downward (if a < 0).

Graph of linear polynomial y=2x-4 showing a straight line crossing the x-axis at the zero point x=2, with labeled axes and y-intercept at (0,-4)
🔑 Linear Zero Formula
For any linear polynomial ax + b (where a ≠ 0):
Zero = x = −b/a = −(constant term) ÷ (coefficient of x)
Example: Zero of 4x − 20 = −(−20)/4 = 20/4 = 5

Quadratic Polynomials: 3 Cases for the Parabola

For y = ax² + bx + c, the parabola opens upward when a > 0 and downward when a < 0. Where it meets the x-axis determines the number of zeroes.

Three cases of quadratic polynomial graphs: Case 1 parabola crosses x-axis at two distinct points alpha and beta, Case 2 parabola touches x-axis at exactly one repeated zero, Case 3 parabola stays above x-axis with no real zeroes

The Three Cases Explained

✅ Case 1: Two Distinct Zeroes
Parabola crosses x-axis at two different points.
There are two different real zeroes: α and β.
e.g. x² − 5x + 6 → zeroes at 2 and 3
⚠️ Case 2: One Repeated Zero
Parabola just touches the x-axis at one point.
Two equal zeroes: α = β.
e.g. x² − 4x + 4 = (x−2)² → zero at 2 only
❌ Case 3: No Real Zeroes
Parabola doesn’t touch the x-axis at all.
No real zeroes exist.
e.g. x² + x + 1 → always positive, no real root

〰️
Section 5

Graphs of Cubic Polynomials

A cubic polynomial y = ax³ + bx² + cx + d always produces a smooth S-shaped curve. Unlike parabolas, a cubic curve always crosses the x-axis at least once — it must have at least one real zero. It can have 1, 2 (one repeated), or 3 distinct zeroes.

Three graphs of cubic polynomials: y=x³-4x with three distinct zeroes at -2, 0, and 2; y=x³ with one repeated zero at origin; y=x³-x² with zeroes at 0 (double) and 1
📊 Maximum Zeroes Rule
A polynomial of degree n can have at most n zeroes — never more. A degree-4 polynomial has at most 4 zeroes, degree 5 has at most 5, and so on. However, it may have fewer than n zeroes if some are complex (non-real).

α β
Section 6

Relationship Between Zeroes and Coefficients

Here is one of the most useful discoveries in algebra: you don’t need to find the zeroes individually to know their sum and product. These values are directly readable from the coefficients of the polynomial itself!

Formula chart showing sum of zeroes equals negative b over a and product of zeroes equals c over a for quadratic polynomials, plus three similar relationships for cubic polynomials alpha beta gamma

⭐ Core Formulas — Quadratic ax² + bx + c
SUM OF ZEROES (α + β)
−b / a
= −(coefficient of x) ÷ (coefficient of x²)
PRODUCT OF ZEROES (α × β)
c / a
= (constant term) ÷ (coefficient of x²)
WORKED EXAMPLE F
Find the zeroes of x² − 7x + 12 and verify the sum and product relationships
Factorise: x² − 7x + 12 = (x − 3)(x − 4)
Zeroes: x = 3 and x = 4 (call them α = 3, β = 4)

Verify sum:
α + β = 3 + 4 = 7 = −(−7)/1 = −b/a ✓

Verify product:
α × β = 3 × 4 = 12 = 12/1 = c/a ✓

Zeroes: 3 and 4
Sum = 7 = −(−7)/1
Product = 12 = 12/1
WORKED EXAMPLE G
Find the zeroes of 2x² − 9x + 10 and verify both relationships
Factorise using middle-term split:
2x² − 9x + 10 → find two numbers with product = 2 × 10 = 20 and sum = −9
Those numbers: −4 and −5 (since −4 × −5 = 20 and −4 + −5 = −9)
2x² − 4x − 5x + 10 = 2x(x − 2) − 5(x − 2) = (2x − 5)(x − 2)
Zeroes: x = 5/2 and x = 2 (α = 5/2, β = 2)

Verify sum: 5/2 + 2 = 5/2 + 4/2 = 9/2 = −(−9)/2 = −b/a ✓
Verify product: (5/2) × 2 = 5 = 10/2 = c/a ✓

Zeroes: 5/2 and 2
Sum = 9/2 ✓
Product = 5 ✓
WORKED EXAMPLE H
Find the zeroes of x² − 5 without factorising fully
Method: use difference of squares: x² − 5 = (x − √5)(x + √5)
Zeroes: x = √5 and x = −√5 (α = √5, β = −√5)

Verify sum: √5 + (−√5) = 0 = −(0)/1 = −b/a [b = 0 since no x term] ✓
Verify product: √5 × (−√5) = −5 = −5/1 = c/a [c = −5] ✓

Zeroes: √5 and −√5
Sum = 0 ✓
Product = −5 ✓
WORKED EXAMPLE I
Without finding zeroes, find the sum and product of zeroes of 3x² + 11x − 4
Compare with ax² + bx + c: a = 3, b = 11, c = −4

Sum of zeroes = −b/a = −11/3
Product of zeroes = c/a = −4/3

Check by factorising: 3x² + 11x − 4 = (3x − 1)(x + 4)
Zeroes: x = 1/3 and x = −4
Sum: 1/3 + (−4) = 1/3 − 12/3 = −11/3 ✓
Product: (1/3)(−4) = −4/3 ✓

Sum = −11/3
Product = −4/3

⭐ Core Formulas — Cubic ax³ + bx² + cx + d
SUM (α+β+γ)
−b / a
PAIR SUM (αβ+βγ+γα)
c / a
TRIPLE PRODUCT (αβγ)
−d / a
WORKED EXAMPLE J
Verify that 1, 2, and 3 are zeroes of p(x) = x³ − 6x² + 11x − 6, then verify all three relationships
Verify each zero:
p(1) = 1 − 6 + 11 − 6 = 0 ✓
p(2) = 8 − 24 + 22 − 6 = 0 ✓
p(3) = 27 − 54 + 33 − 6 = 0 ✓

Identify coefficients: a=1, b=−6, c=11, d=−6
Sum: α+β+γ = 1+2+3 = 6 = −(−6)/1 = −b/a ✓
Pair sum: αβ+βγ+γα = (1×2)+(2×3)+(3×1) = 2+6+3 = 11 = 11/1 = c/a ✓
Triple product: αβγ = 1×2×3 = 6 = −(−6)/1 = −d/a ✓

Sum = 6 ✓
Pair sum = 11 ✓
Triple = 6 ✓

🧱
Section 7

Constructing a Polynomial from Its Zeroes

We can work backwards: given the zeroes (or just the sum and product of zeroes), we can build the polynomial. For a quadratic, the formula is:

x² − (α + β)x + αβ
= x² − (sum of zeroes)x + (product of zeroes)
Four-step diagram for constructing a polynomial from zeroes, plus three worked examples showing how to use sum and product to build quadratic polynomials
WORKED EXAMPLE K
Build a quadratic polynomial whose zeroes are 4 and −3
Step 1: Find sum = 4 + (−3) = 1
Step 2: Find product = 4 × (−3) = −12
Step 3: Polynomial = x² − (sum)x + (product) = x² − (1)x + (−12)
Result: x² − x − 12

Verify by factoring: x² − x − 12 = (x − 4)(x + 3) → zeroes: 4 and −3 ✓

x² − x − 12
WORKED EXAMPLE L
Find a quadratic with sum of zeroes = −5 and product = 4
Using the formula: x² − (sum)x + (product)
x² − (−5)x + 4 = x² + 5x + 4

Verify: x² + 5x + 4 = (x + 1)(x + 4) → zeroes: −1 and −4
Sum: −1 + (−4) = −5 ✓
Product: (−1)(−4) = 4 ✓

x² + 5x + 4
WORKED EXAMPLE M
Build a quadratic whose zeroes are 2+√3 and 2−√3
Step 1: Sum = (2+√3) + (2−√3) = 4
Step 2: Product = (2+√3)(2−√3) = 4 − 3 = 1 [using (a+b)(a−b) = a²−b²]
Step 3: Polynomial = x² − (4)x + (1) = x² − 4x + 1

x² − 4x + 1

🏋️
Section 8

Practice Exercises with Answers

Set A — Find the Zeroes

Find all zeroes of each polynomial by factorising.

(i) x² − 2x − 15
(ii) 6x² − 7x + 2
(iii) x² − 36
(iv) 2x² + 5x + 3
(v) 4x² − 12x + 9
(vi) x² + 2x − 35
Answers:

(i) 5, −3  |  (ii) 2/3, 1/2  |  (iii) 6, −6  |  (iv) −1, −3/2  |  (v) 3/2, 3/2 (repeated)  |  (vi) 5, −7

Set B — Sum and Product of Zeroes

Find the zeroes and verify the sum and product relationships.

(i) 3x² − 10x + 8
(ii) 5x² + 13x + 6
(iii) x² − 2√5 x + 5
(iv) 4x² − 8x
(v) 7x² − 28
(vi) 3x² − x − 10
Answers:

(i) 2, 4/3; Sum=10/3=−(−10)/3 ✓, Prod=8/3 ✓  |  (ii) −2, −3/5; Sum=−13/5 ✓, Prod=6/5 ✓
(iii) √5, √5; Sum=2√5 ✓, Prod=5 ✓  |  (iv) 0, 2; Sum=2=8/4 ✓, Prod=0 ✓
(v) 2, −2; Sum=0=0/7 ✓, Prod=−4=−28/7 ✓  |  (vi) 2, −5/3; Sum=1/3 ✓, Prod=−10/3 ✓

Set C — Construct the Polynomial

Write a quadratic polynomial for each sum and product of zeroes given.

(i) Sum = 3, Product = −10
(ii) Sum = 0, Product = −7
(iii) Sum = −1/4, Product = −1
(iv) Sum = √3, Product = 1/3
(v) Sum = 4, Product = 4
(vi) Sum = −6, Product = 8
Answers:

(i) x² − 3x − 10  |  (ii) x² − 7  |  (iii) x² + x/4 − 1 or 4x² + x − 4
(iv) x² − √3 x + 1/3  |  (v) x² − 4x + 4 = (x−2)²  |  (vi) x² + 6x + 8

Set D — Mixed Challenge Problems

(1) If one zero of x² − 13x + k is 4, find k and the other zero.
(2) If α and β are zeroes of 2x² − 7x + 3, find the value of α² + β² (use: α²+β² = (α+β)² − 2αβ).
(3) The zeroes of a quadratic are in ratio 3:4. If their sum is 14, find the quadratic polynomial.
(4) If −2 is a zero of p(x) = 3x² + kx − 10, find k.
(5) A quadratic polynomial has zeroes whose sum is 1 and whose sum of squares is 13. Find the polynomial.
(6) For the cubic x³ − 6x² + 11x − 6 with zeroes 1, 2, 3: verify all three coefficient relationships.
Answers:

(1) p(4)=0 → 16−52+k=0 → k=36; other zero=13−4=9  |  (2) (7/2)²−2(3/2)=49/4−3=37/4
(3) Ratio 3:4 → zeroes 6,8; product=48; poly: x²−14x+48  |  (4) p(−2)=0 → 12−2k−10=0 → k=1
(5) α+β=1, α²+β²=13 → αβ=(1−13)/2=−6; poly: x²−x−6  |  (6) Sum=6=6/1 ✓, Pair=11 ✓, Prod=6 ✓

📌
Section 9

Chapter Summary

🔑 Everything on One Page

📐 TYPES OF POLYNOMIALS
Degree 1 = Linear (ax+b)
Degree 2 = Quadratic (ax²+bx+c)
Degree 3 = Cubic (ax³+bx²+cx+d)
Degree n = at most n zeroes
0️⃣ ZEROES
k is a zero if p(k) = 0
Geometrically: x-intercepts of y=p(x)
Linear: exactly 1 zero
Quadratic: 0, 1, or 2 zeroes
α β QUADRATIC FORMULAS
Sum α+β = −b/a
Product αβ = c/a
Build: x² − (α+β)x + αβ
α β γ CUBIC FORMULAS
Sum = −b/a
Pair sum αβ+βγ+γα = c/a
Triple product αβγ = −d/a
📈 PARABOLA SHAPES
a > 0 → opens upward ∪
a < 0 → opens downward ∩
Zeroes = where curve meets x-axis
🧱 CONSTRUCTING POLYNOMIALS
Given sum S and product P:
Polynomial = x² − Sx + P
General form: k(x² − Sx + P)

✅ Quick-Check: 8 Things to Remember

1
The degree of a polynomial is the highest power of the variable. It controls everything: how many zeroes exist, the shape of the graph.
2
p(k) = 0 means k is a zero. Always substitute and check — this is the definition that unlocks everything.
3
Zeroes = x-intercepts of y = p(x). They are visible on the graph. Count where the curve meets the x-axis.
4
A quadratic parabola has at most 2 zeroes. It may cross, touch, or miss the x-axis entirely.
5
Sum of zeroes of ax²+bx+c = −b/a. Product = c/a. You can read these directly without solving.
6
For a cubic ax³+bx²+cx+d: sum = −b/a, pair-product sum = c/a, triple product = −d/a.
7
To build a quadratic from zeroes: p(x) = x² − (sum)x + (product). Simple and powerful.
8
A polynomial of degree n has AT MOST n zeroes — never more, but could have fewer if roots are complex.
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