Polynomials · Lesson 4 of 4
Chapter Summary and Practice
“The grand finale where we wrap up all the polynomial drama and give you a chance to prove you're the master of the variables.”
• Revise the main ideas of the chapter. • Identify and classify polynomials correctly. • Interpret zeroes algebraically and geometrically. • Use the relationship between zeroes and coefficients. • Form polynomials from given zeroes. • Practise exam-style questions independently.
Chapter Summary: Polynomials
In this chapter, we studied what polynomials are, how their zeroes appear on graphs, and how those zeroes are related to the coefficients of quadratic and cubic polynomials. Use this lesson to revise the full chapter and test your understanding.
A polynomial can be studied in three connected ways: • Algebraically — by examining its terms, degree and values. • Geometrically — by studying its graph and x-intercepts. • Structurally — by connecting its zeroes with its coefficients.
1. Introduction to Polynomials
A polynomial in one variable is an algebraic expression in which the powers of the variable are non-negative whole numbers.
| Expression | Classification | Reason |
|---|---|---|
| 3x² − 5x + 2 | Polynomial | Powers are 2, 1 and 0 |
| 7 | Polynomial | It can be written as 7x⁰ |
| 1/x + 2 | Not a polynomial | x has power −1 |
| √x + 1 | Not a polynomial | x has power 1/2 |
| 2ˣ + 3 | Not a polynomial | The variable is in the exponent |
For p(x) = 4x³ − 3x² + 2x − 7: • Variable: x • Terms: 4x³, −3x², 2x and −7 • Coefficients: 4, −3 and 2 • Constant term: −7 • Degree: 3
| Type | Meaning | Example |
|---|---|---|
| Constant | 0 | 5 |
| Linear | 1 | 2x − 3 |
| Quadratic | 2 | x² + 4x + 1 |
| Cubic | 3 | x³ − 2x |
| Monomial | One term | 7x² |
| Binomial | Two terms | x² − 9 |
| Trinomial | Three terms | x² + 5x + 6 |
The degree is the highest power of the variable with a non-zero coefficient. It is not the largest coefficient.
2. Zeroes of a Polynomial
A real number k is called a zero of p(x) if p(k) = 0.
Problem
Check whether 2 is a zero of p(x) = x² − 5x + 6.
- 1.Substitute x = 2.
- 2.p(2) = 2² − 5(2) + 6.
- 3.p(2) = 4 − 10 + 6 = 0.
- 4.Therefore, 2 is a zero of the polynomial.
3. Geometrical Meaning of Zeroes
The zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) intersects or touches the x-axis.
| Polynomial | Graph | Possible Number of Zeroes |
|---|---|---|
| Linear polynomial | Straight line | Exactly 1 zero |
| Quadratic polynomial | Parabola | 0, 1 or 2 real zeroes |
| Cubic polynomial | Cubic curve | At most 3 real zeroes |
| Polynomial of degree n | Varies | At most n zeroes |
A graph does not need to cross the x-axis. If it only touches the x-axis and turns back, the point of contact still represents a zero.
4. Relationship Between Zeroes and Coefficients
Sum of zeroes = negative coefficient of x ÷ coefficient of x². Product of zeroes = constant term ÷ coefficient of x².
5. Forming a Polynomial from Its Zeroes
Problem
For p(x) = x² − 7x + 12, find its zeroes, verify their relationship with the coefficients, and state the x-intercepts.
- 1.Factorise: x² − 7x + 12 = (x − 3)(x − 4).
- 2.Therefore, the zeroes are 3 and 4.
- 3.Their sum is 3 + 4 = 7.
- 4.Using coefficients: −b/a = −(−7)/1 = 7.
- 5.Their product is 3 × 4 = 12.
- 6.Using coefficients: c/a = 12/1 = 12.
- 7.The graph intersects the x-axis at (3, 0) and (4, 0).
Key Points
- Write the polynomial in descending powers of x.
- Identify every coefficient with its correct sign.
- Insert zero for the coefficient of a missing term.
- Use p(k) = 0 when checking whether k is a zero.
- Count x-intercepts when reading zeroes from a graph.
- Choose the quadratic or cubic coefficient formula carefully.
- Show all substitution and simplification steps.
- Verify your result whenever factorisation is easy.
• Treating expressions with negative or fractional powers as polynomials. • Confusing the degree with the coefficient. • Counting y-intercepts instead of x-intercepts. • Forgetting that touching the x-axis also represents a zero. • Using b/a instead of −b/a for the sum of zeroes. • Ignoring the sign of a negative coefficient. • Forgetting zero coefficients for missing terms. • Assuming a polynomial of degree n must have exactly n real zeroes.
Practice A: Basic Concepts
Practice Problems
- State whether 5x³ − 2x + 7 is a polynomial. Give a reason.
- State whether 3/x + x² is a polynomial. Give a reason.
- Identify the terms, coefficients, constant term and degree of 6x⁴ − 3x² + 8.
- Classify 4x³ − 5x according to degree and number of terms.
- Classify x² + 3x + 2 according to degree and number of terms.
- Find p(−2) when p(x) = x² + 3x − 4.
- Check whether 3 is a zero of x² − 5x + 6.
- Find the zero of the linear polynomial 4x − 20.
Practice B: Graphs and Zeroes
- A graph intersects the x-axis at x = −2 and x = 5. Write its zeroes.
- A parabola touches the x-axis at (4, 0). How many distinct zeroes does it have?
- A quadratic graph remains completely above the x-axis. How many real zeroes does it have?
- A cubic graph intersects the x-axis at three points. What can you say about its number of real zeroes?
- Can a quadratic polynomial have three distinct zeroes? Explain.
- Can a polynomial of degree 4 have five distinct zeroes? Explain.
- The graph of a polynomial touches the x-axis once and crosses it twice. How many distinct zeroes does it have?
- Find the x-intercept of y = 3x + 6.
Practice C: Zeroes and Coefficients
- Find the sum and product of the zeroes of x² − 9x + 20.
- Find the sum and product of the zeroes of 2x² + 5x − 3.
- Find the sum and product of the zeroes of 4x² − 16.
- Verify the relationship between zeroes and coefficients for x² − x − 6.
- Form a quadratic polynomial whose zeroes are 5 and −2.
- Form a quadratic polynomial whose sum of zeroes is 8 and product is 15.
- Form a polynomial whose zeroes are 1/2 and −3.
- For 2x³ − 5x² − 4x + 3, find the sum of zeroes, sum of pairwise products and product of zeroes.
Mixed Challenge Questions
- If one zero of x² − 7x + 10 is 2, find the other zero using the sum of zeroes.
- If the zeroes of 3x² − 8x + k have product 2, find k.
- If the sum of the zeroes of 2x² + kx − 6 is 5, find k.
- Find a quadratic polynomial whose zeroes are the reciprocals of 2 and 3.
- The zeroes of a quadratic polynomial have sum −4 and product −5. Form the polynomial.
- A quadratic polynomial has zeroes α and β such that α + β = 6 and αβ = 8. Find α² + β².
- For a cubic polynomial, α + β + γ = 4, αβ + βγ + γα = −1 and αβγ = −6. Form the simplest cubic polynomial.
- Explain why x² + 4 has no real zeroes using its graph.
Problem
If one zero of x² − 7x + 10 is 2, find the other zero without factorising.
- 1.Let the other zero be β.
- 2.For x² − 7x + 10, sum of zeroes = −b/a = 7.
- 3.Therefore, 2 + β = 7.
- 4.β = 5.
- 5.Hence, the other zero is 5.
Problem
The zeroes α and β of a quadratic polynomial satisfy α + β = 6 and αβ = 8. Find α² + β².
- 1.Use the identity α² + β² = (α + β)² − 2αβ.
- 2.Substitute α + β = 6 and αβ = 8.
- 3.α² + β² = 6² − 2(8).
- 4.α² + β² = 36 − 16 = 20.
Quiz
Which expression is a polynomial?
What is the degree of 7x⁵ − 3x² + 1?
What represent the zeroes of a polynomial on its graph?
How many real zeroes can a quadratic polynomial have?
For ax² + bx + c, what is the sum of the zeroes?
What is the product of the zeroes of x² − 5x + 6?
Which polynomial has zeroes 3 and −2?
For ax³ + bx² + cx + d, what is αβγ?
Self-Assessment Checklist
Before moving to the next chapter, check whether you can: • Recognise whether an expression is a polynomial. • Identify terms, coefficients, constant term and degree. • Classify linear, quadratic and cubic polynomials. • Check whether a number is a zero. • Read the number of zeroes from a graph. • Use the quadratic and cubic coefficient relationships. • Form a polynomial from given zeroes or from their sum and product. • Solve mixed questions without looking at the formulas.
• A polynomial contains only non-negative whole-number powers of its variable. • A zero k satisfies p(k) = 0. • Graphically, zeroes are the x-coordinates where y = p(x) meets the x-axis. • A polynomial of degree n has at most n zeroes. • For ax² + bx + c: α + β = −b/a and αβ = c/a. • For ax³ + bx² + cx + d: α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a. • A quadratic polynomial with sum S and product P can be written as x² − Sx + P.