Question

x-a) is a factor of x 3 – mx 2

-2nax + na 2

, then prove that a= m +n and a≠0

Answer 1

Hope u like my process
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Let f(x) = x³ - mx² - 2nax +na²

Now..if (x-a) is a factor of x. Then,

f(x) = 0, when x= a

So,
$= > {x}^{3} - m {x}^{2} - 2nax + n {a}^{2} = f(x) \\ \\ or. \: \: {(a)}^{3} - m {(a)}^{2} - 2na \times a + n {a}^{2} = 0 \\ \\ or. \: \: {a}^{3 } - m {a}^{2} - 2n {a}^{2} + n {a}^{2} = 0 \\ \\ or. \: \: {a}^{2} (a - m - n) = 0 \\ \\ or. \: \: a - m - n = 0 \\ < since \: \: \: a \: \: not \: equals \: \: 0 > \\ ........................................... \\ \\ or. \: \: a = m + n.... < proved > ...$
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Answer 2

Step-by-step explanation:

$\bf \underline{Solution-}$

Let p(x) = x³ - mx² - 2nax + na².

If (x - a) is a factor of p(x), then p(a) = 0

Now, p(a) = a³ - m(a)² - 2na(a) + na² = 0

⇒a³ - a²m - 2a²n + na² = 0

⇒. a³ - a²m - a²n = 0

⇒ a²(a - m - n) = 0

Since, a ≠ 0

∴ a - m - n = 0

⇒ a = m + n

Hence, proved.