Find the zeroes of the quadratic polynomial 4×2 – 4x – 3 and verify the relationship between the zeroes and the coefficients of the polynomial.
By BYJU'S Exam Prep
Updated on: October 17th, 2023
The zeroes of the polynomial 4x2 – 4x – 3 are 3/2 and -½. The word “polynomial” is made up of the words “nominal,” which means “terms,” and “poly,” which means “many.” A polynomial (No division operation by a variable) is an expression made up of variables, constants, and exponents that are combined using mathematical operations including addition, subtraction, multiplication, and division. Depending on how many terms are included, the expression is classified as a monomial, binomial, or trinomial.
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f(x) = 4×2 – 4x – 3. Find the Zeroes and Verify the Relationship between the Zeroes and the Coefficients.
The question states Find the zeroes of the quadratic polynomial 4×2-4x-3 and verify the relationship between the zeroes and the coefficients of the polynomial.” The zeros of the following quadratic polynomials 4×2 – 4x – 3 are 3/2 and -½ and the relationship between the zeroes and the coefficients of the polynomial is verified in the below steps. Follow these three steps to find the zeros of the following quadratic polynomials 4×2 – 4x – 3:
Step 1: Form the equation
The given polynomial is 4×2 – 4x – 3
We are aware that a polynomial’s zeroes are evaluated by equating them to zero.
p(s) = 0
Hence the above equation becomes 4×2 – 4x – 3 = 0.
Step 2: Solve to find the zeroes
4×2 – 4x – 3 = 0
The above equation can be written as:
4×2 – 6x + 2x – 3 = 0
2x (2x – 3) + 1 (2x – 3) = 0
(2x + 1)(2x – 3) = 0
Hence, x = 3/2 and -½
Step 3: Verification
We know that for a given polynomial as2 + bs + c
Sum of the zeroes = -b/a and product of the zeroes = c/a
Substituting the values in the above formula we get:
Sum of the zeroes = 3/2 + -½ = 1
Again, -b/a = – (-4)/4 = 1
Product of the zeroes = 3/2 * -½ = -¾
Again, c/a = -¾
Thus, the relationship between the zeroes and the coefficients of the polynomial 4×2 – 4x – 3 is verified. Hence, the zeroes of this polynomial are 3/2 and -½.
Summary:
The zeros of the quadratic polynomial 4×2 – 4x – 3 are 3/2, -½, and the relationship between the zeroes and the coefficients of the polynomial is verified. By substituting values in the sum of the zeroes = -b/a and the product of the roots = c/a, we can easily find the zeros of the quadratic polynomial 4×2 – 4x – 3.
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