Question

find the value of k if (x-1) is a factor of 4x^3+3x^2-4x+k​

Answer 1

$\sf\underline\red{Given:-}$

$\sf Equation \:\: \longrightarrow 4x³+3x²-4x+k$

$\sf (x-1) \:\: is \:\: a \:\: factor$

$\sf\underline\red{To \:\: Find:-}$

$\sf Value \:\: of \:\: k=?$

$\sf\underline\red{Solution:-}$

$\sf (x-1)=0$

$\sf x=1$

$\sf Substituting \:\: x \:\: in \:\: the \:\: equation$

$\sf 4(1)³+3(1)²-4(1)+k=0$

$\sf 4+3-4+k=0$

$\sf k=-3$

$\sf\green{ Value \:\: of \:\: k \:\: is \:\: -3 }$

Answer 2

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Let p(x) = 4x³ + 3x² - 4x + k

& g(x) = (x - 1)

Then, g(x) = 0

⇒ (x - 1) = 0

⇒ x = 0 + 1

⇒ x = 1

Now, By factor theorem

if (x - 1) is a factor of p(x) then p(α) = 0

So, p(α) = 0 ⇒ 4(1)³ + 3(1)² - 4(1) + k = 0

⇒ 4 × 1 + 3 × 1 - 4 × 1 + k = 0

⇒ 4 + 3 - 4 + k = 0

⇒ 7 - 4 + k = 0

⇒ 3 + k = 0

⇒ k = 0 - 3

⇒ k = - 3

∴ The required value of k is - 3.

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