How do you write the equation of a quadratic polynomial in standard form?

The equation of a quadratic polynomial is written in standard form as p(x) = ax² + bx + c, where 'a', 'b', and 'c' are real numbers, and 'a' is not equal to 0. This form arranges the terms in descending order of the exponent of the variable 'x'.

Understanding the Components

Here's a breakdown of each component in the standard form:

• ax²: This is the quadratic term. 'a' is the coefficient of x² and determines the parabola's direction (upward if a > 0, downward if a < 0) and how "wide" or "narrow" it is. The reference states that a ≠ 0, because if a = 0, the expression becomes linear, not quadratic.

• bx: This is the linear term. 'b' is the coefficient of 'x' and affects the position of the parabola's vertex.

• c: This is the constant term. 'c' represents the y-intercept of the parabola (the point where the parabola crosses the y-axis).

Steps to Write a Quadratic Polynomial in Standard Form:

1. Identify the Terms: Look for terms with x², x, and a constant term.
2. Arrange the Terms: Order the terms in descending order of their exponents: first the x² term, then the x term, and finally the constant term.
3. Combine Like Terms: If there are multiple terms with the same exponent, combine them.
4. Ensure 'a' is not Zero: The coefficient of the x² term (a) must not be zero for it to be a quadratic.
5. Express as p(x): Write the expression as p(x) = ax² + bx + c

Examples

Here are a few examples to illustrate writing quadratic polynomials in standard form:

• Example 1:

  • Original: 3x + 2x^2 - 5
  • Standard Form: p(x) = 2x^2 + 3x - 5

• Example 2:

  • Original: 7 - x^2 + 4x
  • Standard Form: p(x) = -x^2 + 4x + 7

• Example 3:

  • Original: x(x - 2) + 1
  • Expand and Simplify: x^2 - 2x + 1
  • Standard Form: p(x) = x^2 - 2x + 1

In each example, the goal is to rearrange the terms so that the highest power of x comes first, followed by the lower powers, and ending with the constant term.

Importance of Standard Form

Writing a quadratic polynomial in standard form makes it easier to:

• Identify coefficients: Easily identify the values of a, b, and c.
• Solve quadratic equations: The quadratic formula relies on the standard form.
• Graph parabolas: Understanding the coefficients helps in sketching the graph.