If α and β are the zeros of the quadratic polynomial f(x) = x2 – 1, find a quadratic polynomial whose zeros are 2α/β and 2β/α.
By BYJU'S Exam Prep
Updated on: October 17th, 2023
If α and β are the zeros of the quadratic polynomial f(x) = x2 – 1, the quadratic polynomial whose zeros are 2α/β and 2β/α is f(x)=(x2+4x+4). Before attempting to answer a polynomial equation, candidates should express it in standard form. Factor it, then when each variable factor reaches zero, set them all to zero. The answers to the derived equations are the solutions to the original equations.
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f(x) = x2 – 1, Find a Quadratic Polynomial whose Zeros are 2α/β and 2β/α.
The question states If α and β are the zeros of the quadratic polynomial f(x) = x2 – 1, find a quadratic polynomial whose zeros are 2α/β and 2β/α.” A polynomial may have some instances in which the variable’s value is zero. Polynomial zero describes these numbers. They are also occasionally known as polynomial roots. We frequently identify the zeros of quadratic equations in order to obtain the answers to the provided problems. A polynomial answer can be explained in one of two ways:
- Solving Linear Polynomials
- Solving Quadratic Polynomials
Given, f(x) = x2−1 = 0
(x-1)(x+1) = 0
x = 1, -1
Hence, α = 1, β = -1
Now, the new roots are 2α/β and 2β/α
2α/β = -2 and 2β/α = -2
Sum of new roots = -4
Product of new roots = 4
Hence, the required quadratic polynomial is f(x)=(x2+4x+4).
Summary:
If α and β are the zeros of the quadratic polynomial f(x) = x2 – 1, find a quadratic polynomial whose zeros are 2α/β and 2β/α.
The quadratic polynomial whose zeros are 2α/β and 2β/α is f(x)=(x2+4x+4), if α and β are the zeros of the quadratic polynomial f(x) = x2 – 1. Division, subtraction, multiplication, and addition are polynomial operations. Any polynomial can be easily solved using fundamental algebraic ideas and factorization strategies. Setting the right-hand side of the polynomial equation equal to zero is the first step in solving it.
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