In algebra, a quadratic polynomial is a polynomial of degree 2. It has the form:
[f(x) = ax^2 + bx + c]
where (a), (b), and (c) are coefficients, and (a \neq 0). Quadratic polynomials can be represented in the standard form, vertex form, or factored form. When a quadratic polynomial is given in factored form with its zeroes, it becomes convenient to find the equation of the polynomial.
Given that the zeroes of a quadratic polynomial are -3 and 4, we can write the factors of the polynomial as:
[(x + 3)(x – 4)]
Expanding this expression gives us the quadratic polynomial in factored form:
[f(x) = (x + 3)(x – 4)]
Now, let’s expand this expression to find the quadratic polynomial in standard form:
[f(x) = x^2 – 4x + 3x – 12]
[f(x) = x^2 – x – 12]
So, the quadratic polynomial with zeroes -3 and 4 is: (f(x) = x^2 – x – 12).
Understanding Quadratic Polynomials:
Quadratic polynomials play a significant role in algebra and have several key characteristics. Here are some key concepts related to quadratic polynomials:
1. Vertex Form:
The vertex form of a quadratic polynomial is given as:
[f(x) = a(x – h)^2 + k]
Where (h, k) represents the vertex of the parabola.
2. Standard Form:
The standard form of a quadratic polynomial is:
[f(x) = ax^2 + bx + c]
Where ‘a’ determines the direction and width of the parabola, and the vertex is located at the point ((-b/2a, f(-b/2a))).
3. Axis of Symmetry:
The axis of symmetry of a parabola given by a quadratic function is the vertical line passing through the vertex.
4. Discriminant:
The discriminant of a quadratic polynomial (ax^2 + bx + c) is given by (\Delta = b^2 – 4ac). It helps determine the nature of the roots of the quadratic equation.
Finding the Quadratic Polynomial with Given Zeroes:
When provided with the zeroes of a quadratic polynomial, solving for the polynomial involves understanding the relationship between the roots and the factors. The zeros of a function are the points where the function equals zero. In the case of a quadratic polynomial, they represent the points where the parabola intersects the x-axis.
To find the quadratic polynomial with zeroes -3 and 4, we utilized the concept that if (x = a) is a zero of a function, then ((x – a)) is a factor of the function. By multiplying these factors together, we can determine the quadratic polynomial that satisfies these conditions.
Key Takeaways:
- Quadratic polynomials are polynomials of degree 2.
- The zeroes of a quadratic polynomial are the points where the polynomial intersects the x-axis.
- The relationship between the zeroes and factors of a quadratic polynomial helps in determining the polynomial with given zeroes.
Frequently Asked Questions (FAQs):
1. What are the properties of a quadratic polynomial?
Quadratic polynomials have a degree of 2, leading coefficient ‘a’ (where (a \neq 0)), and typically graph as a parabola.
2. How do you find the zeroes of a quadratic polynomial?
To find the zeroes of a quadratic polynomial, set the polynomial equal to zero and solve the resulting quadratic equation using factoring, the quadratic formula, or completing the square.
3. What is the significance of the discriminant in quadratic polynomials?
The discriminant ((b^2 – 4ac)) determines the nature of the roots of a quadratic equation. If the discriminant is positive, there are two distinct real roots; if it’s zero, there’s one real root; and if it’s negative, there are two complex roots.
4. How does the vertex form differ from the standard form of a quadratic polynomial?
The vertex form (f(x) = a(x – h)^2 + k) is written to easily identify the vertex of the parabola at the point (h, k), while the standard form (f(x) = ax^2 + bx + c) represents the polynomial in a more general format.
5. Can a quadratic polynomial have imaginary zeroes?
Yes, a quadratic polynomial can have imaginary zeroes. This will be the case if the discriminant is negative, resulting in complex roots.
Understanding the properties and methods for finding the polynomial based on its zeroes enhances your grasp of quadratic functions and their graphical representation. By utilizing the relationship between the roots and factors of a quadratic polynomial, we can efficiently determine the desired polynomial with the given zeroes.