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2.3 - Relationship between Zeroes and Coefficients of a Polynomial

Interactive Audio Lesson

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Zeroes of Quadratic Polynomials

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Teacher
Teacher

Let's begin our discussion on quadratic polynomials. Can anyone remind me what a quadratic polynomial looks like?

Student 1
Student 1

Is it in the form `ax^2 + bx + c`?

Teacher
Teacher

Exactly! Now, if we have a quadratic polynomial like `p(x) = 2x^2 - 8x + 6`, how do we find its zeroes?

Student 2
Student 2

We can factor it or use the quadratic formula.

Student 3
Student 3

Are the zeroes related to the coefficients?

Teacher
Teacher

Great question! The sum of the zeroes, `α + β`, is given by the formula `-b/a`, and the product, `αβ`, is given by `c/a`. Let's calculate these for our polynomial.

Student 4
Student 4

So for `p(x)`, `-(-8)/2 = 4` is the sum, and `6/2 = 3` is the product!

Teacher
Teacher

Yes, exactly! So the zeroes here, `1` and `3`, satisfy both conditions. Let's summarize this concept.

Teacher
Teacher

"Remember: For any quadratic polynomial `ax^2 + bx + c`, the relationships between zeroes and coefficients are:

Examples of Quadratic Polynomial Relationships

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Teacher
Teacher

Let's move on to an example. Consider the polynomial `p(x) = 3x^2 + 5x - 2`. How can we find its zeroes?

Student 1
Student 1

We can factor it into `(3x - 1)(x + 2)` and set each factor to zero.

Teacher
Teacher

Correct! And what are the zeroes then?

Student 2
Student 2

The zeroes are `1/3` and `-2`.

Teacher
Teacher

Now let's check the relationships. What do we see for the sum and product?

Student 3
Student 3

Sum: `1/3 - 2 = -5/3`, which equals `-5/3` indeed, and product: `(1/3)*(-2) = -2/3`, which fits too!

Teacher
Teacher

Fantastic! Keep practicing these relationships, as they are critical in future sections.

Cubic Polynomials

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Teacher
Teacher

Moving on, let’s talk about cubic polynomials represented as `p(x) = ax^3 + bx^2 + cx + d`. Do we have a similar relationship?

Student 2
Student 2

Yes! The sum of the zeroes is still related to the coefficients.

Teacher
Teacher

"Exactly! The relationships for zeroes `α, β, γ` become:

Review and Reinforce

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Teacher
Teacher

Before we wrap up, let's review! What can we say about zeroes in terms of graphical representations?

Student 3
Student 3

The zeroes represent x-coordinates where the graph intersects the x-axis.

Teacher
Teacher

Absolutely! This is critical for understanding polynomial behaviors. And how many zeroes can a cubic polynomial have?

Student 2
Student 2

At most three zeroes!

Teacher
Teacher

"Excellent! Lastly, let's remember the formulas for any quadratic polynomial relation:

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

The section explains the relationship between the zeroes of a polynomial and its coefficients, particularly focusing on quadratic and cubic polynomials.

Standard

This section delves into how the zeroes of quadratic and cubic polynomials are influenced by their coefficients, establishing important formulas for their sums and products. Examples illustrate these relationships through practical calculations.

Detailed

Relationship between Zeroes and Coefficients of a Polynomial

In this section, we explore the profound relationship between the zeroes of polynomial functions and their coefficients. We start with quadratic polynomials, represented in the general form as p(x) = ax^2 + bx + c, where a, b, and c are real coefficients with a ≠ 0.

Key Concepts:

  • Zeroes of Quadratic Polynomials: The zeroes can be found through factoring, and specific relationships define how they relate to coefficients:
  • Sum of Zeroes:
    • Formula: α + β = -b/a
  • Product of Zeroes:
    • Formula: α * β = c/a

We examine several examples:
- Example 1: For p(x) = 2x^2 - 8x + 6, the zeroes 1 and 3 can be derived; their sum and product verify the aforementioned formulas.
- Example 2: For p(x) = 3x^2 + 5x - 2, zeroes derived (1/3 and -2) also fit the patterns established.

Next, we extend to cubic polynomials of the general form p(x) = ax^3 + bx^2 + cx + d and establish relationships:
- Sum of the Zeroes:
- α + β + γ = -b/a
- Sum of Products of Zeroes Taken Two at a Time:
- αβ + βγ + γα = c/a
- Product of Zeroes:
- αβγ = -d/a

Example 3 demonstrates these principles in action for a cubic polynomial p(x) = 3x^3 - 5x^2 - 11x - 3 where the zeroes 3, -1, -1/3 are verified, showing these relationships hold true.

This section serves as a foundation for understanding polynomials, vital for further algebraic exploration such as polynomial division methods and complex functions.

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Audio Book

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Introduction to Zeroes

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You have already seen that zero of a linear polynomial ax + b is −b/a. We will now try to answer the question raised in Section 2.1 regarding the relationship between zeroes and coefficients of a quadratic polynomial. For this, let us take a quadratic polynomial, say p(x) = 2x² – 8x + 6.

Detailed Explanation

This chunk introduces the concept of how the zeroes of a polynomial relate to its coefficients. In a linear polynomial such as ax + b, the zero is simply found using −b/a. Now we expand this idea to quadratic polynomials, which have a different relationship between their zeroes and coefficients. Using the example polynomial p(x) = 2x² - 8x + 6, we will explore how to find the zeroes and connect them to the coefficients.

Examples & Analogies

Think of the coefficients as ingredients in a recipe. Just like the amount of each ingredient affects the final flavor, the coefficients in a polynomial shape the polynomial's zeroes. If you adjust one ingredient dramatically (like changing the coefficient), it can flip the entire taste profile (the zeroes) of the dish.

Finding Zeroes Through Factorization

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In Class IX, you have learnt how to factorise quadratic polynomials by splitting the middle term. So, here we need to split the middle term ‘– 8x’ as a sum of two terms, whose product is 6 × 2x² = 12x². So, we write 2x² – 8x + 6 = 2x² – 6x – 2x + 6 = 2x(x – 3) – 2(x – 3) = (2x – 2)(x – 3) = 2(x – 1)(x – 3).

Detailed Explanation

In this chunk, we apply the method of splitting the middle term to factor the quadratic polynomial. By rewriting –8x as –6x – 2x, we can factor the polynomial into simpler factors. This step is crucial because finding the zeroes of the function happens when we set this factored form equal to zero. The resulting factors, (x - 1) and (x - 3), directly tell us the zeroes: x = 1 and x = 3.

Examples & Analogies

Imagine you want to find the missing pieces of a puzzle that fit together to form a complete picture. Each piece represents a factor of the polynomial. When we split the middle term, we are essentially breaking down our larger puzzle piece into smaller, manageable pieces—making it easier to see how the full picture comes together (the complete polynomial).

Sum and Product of Zeroes

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Observe that : Sum of its zeroes = 1 + 3 = 4 = −(−8)/2, Product of its zeroes = 1 × 3 = 3 = 6/2.

Detailed Explanation

This chunk highlights the connection between the zeroes and the coefficients of the polynomial p(x). Here, we can calculate both the sum and product of the zeroes derived from our factored form. The sum of the zeroes is shown to equal the negative coefficient of x, divided by the coefficient of x² (which is 2). Similarly, the product of the zeroes matches the constant term divided by the coefficient of x². This reinforces the relationship between the coefficients of the polynomial and its zeroes.

Examples & Analogies

Think of this like a bank account where the total sum of your transactions needs to equal a certain balance. The sum and product of the zeroes are like deposits and withdrawals. If they don't balance out correctly (like if we mess with coefficients), the account won't match our expected total (the graph won't intercept the axes correctly).

General Relationship for Quadratic Polynomials

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In general, if α and β are the zeroes of the quadratic polynomial p(x) = ax² + bx + c, a ≠ 0, then you know that x – α and x – β are the factors of p(x). Therefore, ax² + bx + c = k(x – α)(x – β), where k is a constant.

Detailed Explanation

This part generalizes the relationship established in the previous examples. It states that for any quadratic polynomial, the zeroes (α and β) can be expressed in terms of the coefficients a, b, and c. The factors corresponding to these zeroes become essential in the polynomial's factorized form, reinforcing the idea that the zeroes are intricately linked to the polynomial's coefficients.

Examples & Analogies

Imagine you’re building a bridge with beams. The stability and strength (zeroes) of the bridge depend on how well you arrange these beams (coefficients). Just like the correct configuration of beams gives you a solid, stable structure, the right coefficients will yield the correct zeroes for the polynomial.

Applications to Cubic Polynomials

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Let us consider p(x) = 2x³ – 5x² – 14x + 8. You can check that p(x) = 0 for x = 4, –2, 1/2...

Detailed Explanation

In this chunk, we introduce cubic polynomials and examine the connections between their zeroes and coefficients. We illustrate the process of checking which values produce zero outputs for the polynomial, recognizing that cubic equations can have a maximum of three zeroes. Furthermore, we derive relationships akin to those for quadratic polynomials, such as the sum of the zeroes and the product, extending our understanding.

Examples & Analogies

Consider a car navigating through three different checkpoints (the zeroes). The path it takes (the cubic equation) is influenced by how quickly it can reach each checkpoint (coefficients). The relationships help ensure that the car can reach those destinations seamlessly.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Zeroes of Quadratic Polynomials: The zeroes can be found through factoring, and specific relationships define how they relate to coefficients:

  • Sum of Zeroes:

  • Formula: α + β = -b/a

  • Product of Zeroes:

  • Formula: α * β = c/a

  • We examine several examples:

  • Example 1: For p(x) = 2x^2 - 8x + 6, the zeroes 1 and 3 can be derived; their sum and product verify the aforementioned formulas.

  • Example 2: For p(x) = 3x^2 + 5x - 2, zeroes derived (1/3 and -2) also fit the patterns established.

  • Next, we extend to cubic polynomials of the general form p(x) = ax^3 + bx^2 + cx + d and establish relationships:

  • Sum of the Zeroes:

  • α + β + γ = -b/a

  • Sum of Products of Zeroes Taken Two at a Time:

  • αβ + βγ + γα = c/a

  • Product of Zeroes:

  • αβγ = -d/a

  • Example 3 demonstrates these principles in action for a cubic polynomial p(x) = 3x^3 - 5x^2 - 11x - 3 where the zeroes 3, -1, -1/3 are verified, showing these relationships hold true.

  • This section serves as a foundation for understanding polynomials, vital for further algebraic exploration such as polynomial division methods and complex functions.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • Example of quadratic polynomial: p(x) = 2x^2 - 8x + 6 results in zeroes 1 and 3.

  • Example of cubic polynomial: p(x) = 2x^3 - 5x^2 - 14x + 8 results in zeroes 3, -2, and -1/2.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Zeroes to the left, coefficients right; find their sum, and the product's a delight!

📖 Fascinating Stories

  • Imagine the quadratic fairy who lived in the polynomial forest, her zeroes were found by the magic of coefficients casting their spells.

🧠 Other Memory Gems

  • Remember: S for Sum, P for Product - just link them to b and c!

🎯 Super Acronyms

SPC

  • Sum
  • Product
  • Coefficients - remember these to tackle any polynomial quest.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Zero of a Polynomial

    Definition:

    A value of x such that p(x) = 0 for a polynomial p(x).

  • Term: Quadratic Polynomial

    Definition:

    Polynomial of degree two represented as p(x) = ax^2 + bx + c.

  • Term: Cubic Polynomial

    Definition:

    Polynomial of degree three represented as p(x) = ax^3 + bx^2 + cx + d.

  • Term: Coefficient

    Definition:

    A numerical or constant factor in a term of an algebraic expression.