Forming a Quadratic Polynomial from Zeros

A quadratic polynomial is a polynomial of degree 2. It can be represented in the general form: $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \ne 0$. A key feature of quadratic polynomials is that they have two zeros (also known as roots), which are the values of $x$ that make the polynomial equal to zero. This section focuses on how to construct a quadratic polynomial if you are given the sum and the product of its zeros.

Formulae

If $\alpha$ and $\beta$ are the zeros of a quadratic polynomial, then:

• Sum of the zeros: $\alpha + \beta$
• Product of the zeros: $\alpha \cdot \beta$
• The quadratic polynomial can be formed using: $x^2 – (\text{Sum of zeros})x + (\text{Product of zeros})$
• This translates to: $x^2 – (\alpha + \beta)x + \alpha \beta$

Examples

Example-1: Form a quadratic polynomial whose zeros have a sum of 5 and a product of 6.

Solution: Let $\alpha$ and $\beta$ be the zeros. We are given: $\alpha + \beta = 5$ $\alpha \beta = 6$ The quadratic polynomial is: $x^2 – (\alpha + \beta)x + \alpha \beta$ Substituting the given values: $x^2 – (5)x + 6$ Therefore, the quadratic polynomial is $x^2 – 5x + 6$.

Example-2: Form a quadratic polynomial whose zeros are $\frac{2}{3}$ and $-\frac{1}{2}$.

Solution: Let $\alpha = \frac{2}{3}$ and $\beta = -\frac{1}{2}$. First, find the sum of the zeros: $\alpha + \beta = \frac{2}{3} – \frac{1}{2} = \frac{4 – 3}{6} = \frac{1}{6}$ Next, find the product of the zeros: $\alpha \beta = \frac{2}{3} \cdot (-\frac{1}{2}) = -\frac{1}{3}$ The quadratic polynomial is: $x^2 – (\alpha + \beta)x + \alpha \beta$ Substituting the calculated values: $x^2 – (\frac{1}{6})x – \frac{1}{3}$ Therefore, the quadratic polynomial is $x^2 – \frac{1}{6}x – \frac{1}{3}$.

Common mistakes by students

• Forgetting the negative sign: Students often forget the negative sign in the formula $x^2 – (\text{Sum of zeros})x + (\text{Product of zeros})$. This is a very common mistake.
• Confusing sum and product: Mixing up the sum and product of the zeros when substituting the values into the formula.
• Incorrect arithmetic: Making errors when calculating the sum and/or product of the zeros, especially when dealing with fractions or negative numbers.
• Incorrect coefficients: Failing to simplify the polynomial correctly after substitution, resulting in incorrect coefficients for x terms or the constant term.

Real Life Application

Quadratic equations and polynomials are widely used in many real-life applications, including:

• Physics: The trajectory of a projectile (like a ball thrown in the air) can be modeled using a quadratic equation. Knowing the sum and product of the roots can help analyze the time and location of impact.
• Engineering: Designing bridges, arches, and other structures often involves using quadratic equations to determine the optimal shape and load-bearing capacity.
• Finance: Modeling investment returns and analyzing profit and loss can sometimes utilize quadratic relationships. For example, determining the optimal price of a product to maximize revenue (given a demand function).
• Computer Graphics: Quadratic curves are essential in computer graphics, particularly in creating smooth, realistic-looking shapes and animations.

Fun Fact

The sum and product of the zeros of a quadratic polynomial can be directly related to the coefficients of the polynomial without knowing the actual zeros. For a quadratic polynomial $ax^2 + bx + c$, the sum of the zeros is $-\frac{b}{a}$, and the product of the zeros is $\frac{c}{a}$. This is a consequence of Vieta’s formulas, a powerful tool in polynomial theory.

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