Relationship Between Coefficients and Zeroes

Oct 13, 2025 by windailys.com 19 views

Introduction

In this problem, we are given the sum and product of the zeroes of a quadratic polynomial. We need to find the quadratic polynomial using this information. The sum and product of the zeroes of a quadratic polynomial are related to its coefficients, and we can use this relationship to find the polynomial.

Sum and Product of Zeroes

The sum of the zeroes of a quadratic polynomial ax² + bx + c is given by -b/a, and the product of the zeroes is given by c/a. In this problem, we are given that the sum of the zeroes is 2√3 and the product is 3.

Relationship Between Coefficients and Zeroes

We can use the relationship between the coefficients and the zeroes of a quadratic polynomial to find the polynomial. The sum of the zeroes is -b/a, and the product of the zeroes is c/a. We can write the quadratic polynomial as ax² + bx + c, where a, b, and c are constants.

Finding the Quadratic Polynomial

We are given that the sum of the zeroes is 2√3 and the product is 3. We can write the quadratic polynomial as ax² + bx + c, where a, b, and c are constants. We know that -b/a = 2√3 and c/a = 3. We can use this information to find the values of a, b, and c.

Solving for a, b, and c

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We know that the product of the zeroes is 3, so we can write c/a = 3. We can substitute c = 3a into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: a(x² + x + 3) = a(x² + x + 3).

Finding the Value of a

We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).

Finding the Value of a

We know that -b/a = 2√3, so we can write -b = 2√3a. We can substitute this expression for -b into the quadratic polynomial: a(x² + x + 3).

Simplifying the Quadratic Polynomial

We can simplify the quadratic polynomial by combining

Q&A

Q: What is the sum and product of zeroes of a quadratic polynomial?

A: The sum of the zeroes of a quadratic polynomial ax² + bx + c is given by -b/a, and the product of the zeroes is given by c/a.

Q: How do we find the quadratic polynomial using the sum and product of zeroes?

A: We can use the relationship between the coefficients and the zeroes of a quadratic polynomial to find the polynomial. The sum of the zeroes is -b/a, and the product of the zeroes is c/a. We can write the quadratic polynomial as ax² + bx + c, where a, b, and c are constants.

Q: What is the relationship between the coefficients and the zeroes of a quadratic polynomial?

A: The sum of the zeroes is -b/a, and the product of the zeroes is c/a. We can use this relationship to find the values of a, b, and c.

Q: How do we find the value of a?

A: We can start by solving for a. We know that c/a = 3, so we can write c = 3a. We can substitute this expression for c into the quadratic polynomial: ax² + bx + 3a.

Q: How do we simplify the quadratic polynomial?

A: We can simplify the quadratic polynomial by combining like terms: ax² + bx + 3a = a(x² + x + 3).