Questions: Express (f(x)) in the form (f(x)=(x-k) q(x)+r) for the given value of (k). [ f(x)=x^3+5 x^2+10 x+8, k=-2 ]

Express (f(x)) in the form (f(x)=(x-k) q(x)+r) for the given value of (k). [ f(x)=x^3+5 x^2+10 x+8, k=-2 ]
Transcript text: Express $f(x)$ in the form $f(x)=(x-k) q(x)+r$ for the given value of $k$. \[ f(x)=x^{3}+5 x^{2}+10 x+8, k=-2 \]
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Solution

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Solution Steps

Step 1: Polynomial Division

Divide the polynomial by $(x + 2)$ using synthetic division.

Step 2: Calculations

The quotient polynomial $q(x)$ has coefficients: [1, 7, 24] The remainder is: 56

Step 3: Formulate the Result

Thus, $f(x) = (x+2)(1$x^{3}$ + 7$x^{2}$ + 24$x^{1}$) + 56$

Final Answer:

$f(x) = (x+2)(1$x^{3}$ + 7$x^{2}$ + 24$x^{1}$) + 56$

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