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The form of your answer should either be $p(x)$ or $p(x)+\frac{k}{x-5}$ where $p(x)$ is a polynomial and $k$ is an integer. $\frac{2x^3-11x^2+25}{x-5}$

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Composition of linear and quadratic functions: find an equation

Full solution

Q. The form of your answer should either be

p(x)

or

p(x)+\frac{k}{x-5}

where

p(x)

is a polynomial and

k

is an integer.

\frac{2x^3-11x^2+25}{x-5}

Check Factor Using Synthetic Division: Step $1$ : Check if $x-5$ is a factor of the numerator $2x^3-11x^2+25$ using synthetic division. $\newline$ Divide $2x^3-11x^2+25$ by $x-5$ : $\newline$ - Coefficients of the polynomial: $2$ , $-11$ , $0$ , $25$ $\newline$ - Value for synthetic division: $5$ $\newline$ - Perform synthetic division: $\newline$ $2$ | $2x^3-11x^2+25$ $0$ | $2x^3-11x^2+25$ $1$ | $25$ $\newline$ ------------------------- $\newline$ $2$ | $2x^3-11x^2+25$ $4$ | $2x^3-11x^2+25$ $1$ | $0$ $\newline$ The remainder is $0$ , indicating $x-5$ is a factor.
Write Quotient as Polynomial: Step $2$ : Write the quotient from the synthetic division as a polynomial. $\newline$ The quotient is $2x^2 - x - 5$ .
Simplify Expression: Step $3$ : Since the remainder is $0$ , the expression simplifies to the quotient. $\newline$ The simplified form is $2x^2 - x - 5$ .

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