For what value of k, will the expression (3x^3 – kx^2 + 4x + 16) be divisible by (x – k/2) ? -Maths 9th

For what value of k, will the expression (3x^3 – kx^2 + 4x + 16) be divisible by (x – k/2) ? -Maths 9th
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Answer :

Given f(x) = 3x³ - kx² + 4x + 16. Since (x - k/2) is a factor of polynomial. This means x = k/2 is the zero of the given polynomial. ⇒ f(k/2) = 3(k/2)³ - k(k/2)² + 4(k/2) + 16 ⇒ 0 = 3(k³/8) - 2(k³/4) + 4(k/2) + 16 ⇒ 0 = (3k³ - 2k³ + 16k + 128)/8 ⇒ 0 = 3k³ - 2k³ + 16k + 128 ⇒ 0 = k³ + 16k + 128 ⇒ 0 = k³ - 4k² + 4k² + 32k - 16k + 128 ⇒ 0 = k³ - 4k² + 32k + 4k² - 16k + 128 ⇒ 0 = k(k² - 4k + 32) + 4(k² - 4k + 32) ⇒ 0 = (k + 4)(k² - 4k + 32) ⇒ k = -4.
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