What is the discriminant of the polynomial below?
[tex]4 x^2+4 x+1[/tex]
A. -12
B. 32
C. -4
D. 0
Answer (1)
The discriminant of a quadratic polynomial a x 2 + b x + c is given by b 2 − 4 a c . For the polynomial 4 x 2 + 4 x + 1 :
Identify the coefficients: a = 4 , b = 4 , c = 1 .
Substitute the values into the discriminant formula: D = 4 2 − 4 × 4 × 1 .
Calculate the discriminant: D = 16 − 16 = 0 .
The discriminant of the polynomial is 0 .
Explanation
Understanding the Discriminant We are asked to find the discriminant of the quadratic polynomial 4 x 2 + 4 x + 1 . The discriminant is a value that can be computed from the coefficients of the polynomial and gives information about the nature of the roots of the polynomial. Specifically, for a quadratic polynomial of the form a x 2 + b x + c , the discriminant, denoted by D , is given by the formula D = b 2 − 4 a c .
Identifying Coefficients In the given polynomial, 4 x 2 + 4 x + 1 , we can identify the coefficients as follows:
a = 4 b = 4 c = 1
Calculating the Discriminant Now, we substitute these values into the discriminant formula:
D = b 2 − 4 a c = ( 4 ) 2 − 4 ( 4 ) ( 1 ) = 16 − 16 = 0
Final Answer Therefore, the discriminant of the polynomial 4 x 2 + 4 x + 1 is 0.
Examples
The discriminant is useful in many areas, including determining the nature of the roots of a quadratic equation. For example, when designing a bridge, engineers use quadratic equations to model the curve of the bridge. The discriminant helps them determine if the bridge will intersect the ground (two real roots), touch the ground at one point (one real root), or never touch the ground (no real roots). This ensures the bridge's design is safe and functional.
