Mathematics Class 10 Polynomials Zeroes and Graphs

PREREQUISITES

• Factoring of polynomials
• Graphs of quadratic equations
• Understanding of rational expressions
• Use of quadratic formula
• Basic algebraic operations

MASTER ORGANIZER

FORMULAS & THEOREMS

DIAGRAMS TO KNOW

1. Graph of a Quadratic Equation
2. Name: Graph of a quadratic equation
3. Key features: U-shaped, vertex, axis of symmetry
4. What it represents: The graph of a quadratic equation in the form y = ax^2 + bx + c

5. Common exam focus: Finding the vertex, axis of symmetry, and x-intercepts

6. Number Line

7. Name: Number line
8. Key features: Numbers on a line, distance between numbers
9. What it represents: A visual representation of numbers on a line

10. Common exam focus: Finding the distance between numbers, identifying the midpoint

11. Geometric Figure

12. Name: Geometric figure
13. Key features: Points, lines, angles
14. What it represents: A visual representation of geometric shapes
15. Common exam focus: Identifying geometric shapes, finding the perimeter and area

RAPID REVISION SHEET

• Zeroes of a polynomial are the values of x that make the polynomial equal to zero
• Rational zeroes theorem: if f(x) = 0 has a rational zero, then it must be of the form p/q, where p is a factor of f(0) and q is a factor of f(c)
• Graphs of polynomials can be used to model real-world situations
• Descartes' rule of signs: the number of positive real roots of a polynomial is equal to the number of sign changes in the coefficients of the polynomial or less than it by a multiple of 2
• Conjugate zeros theorem: if a polynomial with real coefficients has a complex zero, then its conjugate is also a zero
• Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
• Rational root theorem: if a rational number p/q is a root of the polynomial f(x), then p is a factor of f(0) and q is a factor of f(c)
• Graph of a quadratic equation is a U-shaped curve
• Number line is a visual representation of numbers on a line
• Geometric figure is a visual representation of geometric shapes

STEP-BY-STEP PROBLEM SOLVER

Problem Type 1: Finding Zeroes of a Polynomial

Problem: Find the zeroes of the polynomial f(x) = x^2 + 4x + 4.

Solution: 1. Factor the polynomial: f(x) = (x + 2)(x + 2) 2. Find the zero: x + 2 = 0, x = -2 3. Check the solution: f(-2) = (-2)^2 + 4(-2) + 4 = 0

Problem Type 2: Graphing a Quadratic Equation

Problem: Graph the quadratic equation y = x^2 + 2x + 1.

Solution: 1. Find the vertex: x = -b / 2a = -2 / 2 = -1 2. Find the y-coordinate of the vertex: y = (-1)^2 + 2(-1) + 1 = 0 3. Find the x-intercepts: x = -b ± √(b^2 - 4ac) / 2a = -2 ± √(4 - 4) / 2 = -2 4. Plot the points: (0, y), (-2, 0), (-1, 0)

Problem Type 3: Finding the Number of Positive Real Roots of a Polynomial

Problem: Use Descartes' rule of signs to find the number of positive real roots of the polynomial f(x) = x^3 - 2x^2 - 5x + 6.

Solution: 1. Count the number of sign changes: 3 2. Since the number of sign changes is odd, the polynomial has one positive real root.

COMMON CONFUSIONS SHEET

A vs B → Explanation Zeroes of a polynomial vs roots of a polynomial → Zeroes of a polynomial are the values of x that make the polynomial equal to zero, while roots of a polynomial are the values of x that make the polynomial equal to zero and are also solutions to the equation.
Conjugate zeros theorem vs rational root theorem → Conjugate zeros theorem states that if a polynomial with real coefficients has a complex zero, then its conjugate is also a zero, while rational root theorem states that if a rational number p/q is a root of the polynomial f(x), then p is a factor of f(0) and q is a factor of f(c).

COMMON MISTAKES & TRAPS

Mistake/Trap → Why it happens → How to avoid Assuming a zero is repeated when it is not → Not checking if the zero is repeated by substituting the value into the polynomial → Check if the zero is repeated by substituting the value into the polynomial.
Not checking the discriminant → Not considering the nature of the solutions → Check the discriminant to determine the nature of the solutions.
Not considering the conjugate zeros → Not considering the complex zeroes → Check for complex zeroes by substituting the value into the polynomial.
Not checking if p and q are factors of f(0) and f(c) → Not considering the rational zeroes → Check if p and q are factors of f(0) and f(c) to determine the rational zeroes.

EXAM ANSWER BUILDER

1-Mark Question

Question: What is the name of the theorem that states that if a polynomial with real coefficients has a complex zero, then its conjugate is also a zero?

Answer: Conjugate zeros theorem.

Key tip: Remember that the conjugate zeros theorem applies to polynomials with real coefficients.

3-Mark Question

Question: Find the solutions to the quadratic equation x^2 + 4x + 4 = 0.

Answer: x = (-b ± √(b^2 - 4ac)) / 2a = (-4 ± √(16 - 16)) / 2 = -2.

Key tip: Use the quadratic formula to find the solutions.

5-Mark Question

Question: Graph the quadratic equation y = x^2 + 2x + 1 and find the x-intercepts.

Answer: First, find the vertex: x = -b / 2a = -2 / 2 = -1. Then, find the y-coordinate of the vertex: y = (-1)^2 + 2(-1) + 1 = 0. Next, find the x-intercepts: x = -b ± √(b^2 - 4ac) / 2a = -2 ± √(4 - 4)