Question

*इनमें से कौन-सा बहुपद पूरी तरह (x - y ) से विभाज्य है? {संकेत: यह बहुपद (x - y) से पूरी तरह विभाज्य होगा अगर उसका एक गुणनखंड (x - y) है।)*

1️⃣ y(x² - y)
2️⃣ x² + y²
3️⃣ x² - xy - x + y
4️⃣ (x - y) - x - y​

Answer 1

Which of these polynomials is completely divisible by (x - y)? {Hint: It will be completely divisible by the polynomial (x - y) if it has a factor (x - y).)

$\textbf{To find:}$

$\textsf{Which of these polynomials is completely divisible by (x - y)? }$

$\mathsf{1.\;y(x^2-y)}$

$\mathsf{2.x^2+y^2}$

$\mathsf{3.\;x^2-xy-x+y}$

$\mathsf{4.\;(x-y)-x-y}$

$\textbf{Solution:}$

$\mathsf{1.\;y(x^2-y)}$

$\textsf{Clearly it is not divisible by x-y}$

$\mathsf{2.x^2+y^2}$

$\textsf{It cannot be written as a prodiuct of factors which containing x-y}$

$\textsf{It is not divisible by x-y}$

$\mathsf{3.\;x^2-xy-x+y}$

$\mathsf{=x(x-y)-1(x-y)}$

$\mathsf{=(x-1)(x-y)}\;\textsf{which is divisble by x-y}$

$\textbf{Answer:}$

$\mathsf{x^2-xy-x+y\;is\;divisible\;by\;x-y}$