Quadratic Polynomial Definition
Quadratic Polynomial Example
Suppose we have a quadratic polynomial x^{2} + 4x + 4 = 0. Then to find the solutions of this equation we factorize it as (x + 2)(x + 2) = 0. Thus, the roots of this quadratic equation will be x = 2, 2.
Quadratic Polynomial Sum and Product of Roots
Common Factor Method
Example: What are the common factors of the terms in the quadratic polynomial equation 8x^{2} − 4x = 0?
Let's apply the distributive law in reverse.
4x is a common factor in the equation.
Thus, 4x(2x  1) are the factors of 8x^{2} − 4x = 0
Sum of Difference Method
Example: Find the solution of (5 + x)(5  x) using the sum of the difference method.
Apply the sum of the difference method for solving the terms.
(a + b)(a  b) = a^{2 } b^{2}
(5 + x)(5  x) = (5^{2}  x^{2}) = 25  x^{2}
Factor By Grouping Method
Example: How can you factorize the quadratic polynomial a^{2}  ac + ab  bc by the grouping method?
a^{2}  ac + ab  bc
Take the common factor from the quadratic polynomial.
= a(a  c) + b(a  c)
= (a  c) (a + b)
Thus, by factoring expressions we get (a  c) (a + b).
Perfect Square Trinomials Method
Example: Is the given quadratic polynomial x^{2 }– 8x + 16 a perfect square?
On using the formula, we get
x^{2} – 8x + 16 = x^{2} – 2(1)(4)x + 4^{2}
= (x  4)^{2}
Thus, the given quadratic polynomial is a perfect square.
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