What is Relation Between Zeroes and Coefficients?

S3-SA1-0794

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The 'zeroes' of a polynomial are the values of the variable that make the polynomial equal to zero. The 'coefficients' are the numbers multiplied by the variables in the polynomial. The relation between them shows how these special values (zeroes) are connected to the numbers (coefficients) in the equation.

Simple Example

Quick Example

Imagine you have a simple math puzzle: x - 5 = 0. The 'zero' here is 5, because if you put 5 in place of x, the equation becomes 5 - 5 = 0. The 'coefficient' of x is 1 (because 1 times x is x), and -5 is also a coefficient (the constant term). The relation connects this 5 to the 1 and -5.

Worked Example

Step-by-Step

Let's take a quadratic polynomial: x^2 - 7x + 10. --- STEP 1: Find the zeroes. We need to find values of x that make x^2 - 7x + 10 = 0. We can factorize this as (x - 2)(x - 5) = 0. --- STEP 2: Identify the zeroes. From (x - 2)(x - 5) = 0, we get x - 2 = 0 or x - 5 = 0. So, the zeroes are x = 2 and x = 5. --- STEP 3: Identify the coefficients. In x^2 - 7x + 10, the coefficient of x^2 is 'a' = 1, the coefficient of x is 'b' = -7, and the constant term is 'c' = 10. --- STEP 4: Check the sum of zeroes. Sum = 2 + 5 = 7. According to the relation, Sum of zeroes = -b/a = -(-7)/1 = 7/1 = 7. This matches! --- STEP 5: Check the product of zeroes. Product = 2 * 5 = 10. According to the relation, Product of zeroes = c/a = 10/1 = 10. This also matches! --- ANSWER: The zeroes (2 and 5) are related to the coefficients (1, -7, 10) by the formulas: Sum of zeroes = -b/a and Product of zeroes = c/a.

Why It Matters

Understanding this helps engineers design systems, data scientists analyze patterns, and economists predict market trends. For instance, in AI/ML, these relations help optimize algorithms for better performance. Many real-world problems can be modeled by polynomials, and finding their zeroes is crucial for solving them, which is used by software developers and researchers.

Common Mistakes

MISTAKE: Forgetting the negative sign in -b/a for the sum of zeroes. | CORRECTION: Always remember the formula is 'minus b over a', so if b is already negative, -b will become positive.

MISTAKE: Mixing up the 'a', 'b', and 'c' values, especially if the polynomial terms are not in order (e.g., 10 - 7x + x^2). | CORRECTION: Always rearrange the polynomial in standard form (ax^2 + bx + c) before identifying a, b, and c.

MISTAKE: Assuming the constant term 'c' is always positive in the product formula (c/a). | CORRECTION: The constant term 'c' can be positive or negative, and its sign must be included correctly in the formula.

Practice Questions

Try It Yourself

QUESTION: For the polynomial x^2 - 5x + 6, find the sum of its zeroes using the relation. | ANSWER: Sum of zeroes = -(-5)/1 = 5

QUESTION: If the zeroes of a quadratic polynomial are 3 and 4, find the product of its zeroes. | ANSWER: Product of zeroes = 3 * 4 = 12

QUESTION: A quadratic polynomial has zeroes -2 and 7. If the coefficient of x^2 is 1, what is the constant term (c) of the polynomial? | ANSWER: Product of zeroes = (-2) * 7 = -14. Since Product = c/a, and a=1, then c = -14.

MCQ

Quick Quiz

For the polynomial 2x^2 + 8x + 6, what is the sum of its zeroes?

-4

-8

The Correct Answer Is:

The sum of zeroes is given by -b/a. Here, a=2 and b=8. So, the sum is -8/2 = -4.

Real World Connection

In the Real World

In cricket analytics, statisticians use polynomial equations to model a batsman's performance over a season. Finding the 'zeroes' could represent points where their form dipped to zero or where a certain target was met. The coefficients help understand the factors (like pitch, opposition) influencing their scores. This helps coaches make strategic decisions.

Key Vocabulary

Key Terms

ZEROES: Values of the variable that make a polynomial equal to zero | COEFFICIENTS: The numerical values multiplied by the variables in a polynomial | QUADRATIC POLYNOMIAL: A polynomial with the highest power of the variable as 2 (e.g., ax^2 + bx + c) | CONSTANT TERM: The term in a polynomial that does not have a variable (e.g., 'c' in ax^2 + bx + c)

What's Next

What to Learn Next

Great job understanding this! Next, you can explore how to form a quadratic polynomial when its zeroes are given. This builds directly on what you've learned about the sum and product of zeroes, helping you create equations from their solutions.