By factor theorem, substituting x = 2 and x = 1/2 both make the polynomial equal 0. Equating the resulting expressions 4p + 10 + r = 0 and p/4 + 5/2 + r = 0 simplifies perfectly to prove p = r.
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By factor theorem, substituting x = 2 and x = 1/2 both make the polynomial equal 0. Equating the resulting expressions 4p + 10 + r = 0 and p/4 + 5/2 + r = 0 simplifies perfectly to prove p = r.
Since the given binomial equations are exact factors, we apply the factor theorem to solve the unknown coefficients. Setting x equal to 2 yields the equation 4p + 10 + r = 0. Setting x equal to 1/2 yields p/4 + 5/2 + r = 0, which multiplies out to p + 10 + 4r = 0. Equating both zero expressions gives 4p + 10 + r = p + 10 + 4r. Cancelling 10 and rearranging the terms simplifies to 3p = 3r, proving p = r.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 4 Exploring Algebraic Identities (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-4/