2.3 - Relationship between Zeroes and Coefficients of a Polynomial
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Zeroes of Quadratic Polynomials
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Let's begin our discussion on quadratic polynomials. Can anyone remind me what a quadratic polynomial looks like?
Is it in the form `ax^2 + bx + c`?
Exactly! Now, if we have a quadratic polynomial like `p(x) = 2x^2 - 8x + 6`, how do we find its zeroes?
We can factor it or use the quadratic formula.
Are the zeroes related to the coefficients?
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.
So for `p(x)`, `-(-8)/2 = 4` is the sum, and `6/2 = 3` is the product!
Yes, exactly! So the zeroes here, `1` and `3`, satisfy both conditions. Let's summarize this concept.
"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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Let's move on to an example. Consider the polynomial `p(x) = 3x^2 + 5x - 2`. How can we find its zeroes?
We can factor it into `(3x - 1)(x + 2)` and set each factor to zero.
Correct! And what are the zeroes then?
The zeroes are `1/3` and `-2`.
Now let's check the relationships. What do we see for the sum and product?
Sum: `1/3 - 2 = -5/3`, which equals `-5/3` indeed, and product: `(1/3)*(-2) = -2/3`, which fits too!
Fantastic! Keep practicing these relationships, as they are critical in future sections.
Cubic Polynomials
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Moving on, let’s talk about cubic polynomials represented as `p(x) = ax^3 + bx^2 + cx + d`. Do we have a similar relationship?
Yes! The sum of the zeroes is still related to the coefficients.
"Exactly! The relationships for zeroes `α, β, γ` become:
Review and Reinforce
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Before we wrap up, let's review! What can we say about zeroes in terms of graphical representations?
The zeroes represent x-coordinates where the graph intersects the x-axis.
Absolutely! This is critical for understanding polynomial behaviors. And how many zeroes can a cubic polynomial have?
At most three zeroes!
"Excellent! Lastly, let's remember the formulas for any quadratic polynomial relation:
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
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
- Formula:
- Product of Zeroes:
- Formula:
α * β = c/a
- Formula:
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.
Youtube Videos
![Relation of Zeroes with Coefficients | Class 10th [ Polynomials ]](https://img.youtube.com/vi/GmOEXDkYqKI/mqdefault.jpg)
Audio Book
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Introduction to Zeroes
Chapter 1 of 5
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Chapter Content
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
Chapter 2 of 5
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Chapter Content
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
Chapter 3 of 5
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Chapter Content
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
Chapter 4 of 5
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Chapter Content
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
Chapter 5 of 5
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Chapter Content
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.
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 zeroes1and3can 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 + dand 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 - 3where the zeroes3, -1, -1/3are 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 & Applications
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
Interactive tools to help you remember key concepts
Rhymes
Zeroes to the left, coefficients right; find their sum, and the product's a delight!
Stories
Imagine the quadratic fairy who lived in the polynomial forest, her zeroes were found by the magic of coefficients casting their spells.
Memory Tools
Remember: S for Sum, P for Product - just link them to b and c!
Acronyms
SPC
Sum
Product
Coefficients - remember these to tackle any polynomial quest.
Flash Cards
Glossary
- Zero of a Polynomial
A value of x such that p(x) = 0 for a polynomial p(x).
- Quadratic Polynomial
Polynomial of degree two represented as p(x) = ax^2 + bx + c.
- Cubic Polynomial
Polynomial of degree three represented as p(x) = ax^3 + bx^2 + cx + d.
- Coefficient
A numerical or constant factor in a term of an algebraic expression.
Reference links
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