Question

If $f\left(x\right)=x^4-12x^3+17x^2-9x+7$, find the value of $f(x+3)$.

Answer 1

$f\left(x\right)=x^4-12x^3+17x^2-9x+7$

$f(x+3)=(x+3{)}^4-12(x+3{)}^3+17(x+3{)}^2-9(x+3)+7$

$=(x^4+(\frac{4}{1})\cdot x^3\cdot 3+(\frac{4}{2})\cdot x^2\cdot 3^2+(\frac{4}{3})\cdot x\cdot 3^3+(\frac{4}{4})\cdot x\cdot 3^4)$

$-12(x^3+(\frac{3}{1})\cdot x^2\cdot 3+(\frac{3}{2})\cdot x\cdot 3^2+(\frac{3}{3})\cdot 3^3)$

$+17(x^2+(\frac{2}{1})\cdot x\cdot 3+(\frac{2}{2})\cdot 3^2)-9(x+3)+7$

$=(x^4+12x^3+54x^2+108x+81)-12(x^3+9x^2+27x+27)+17(x^2+6x+9)-9(x+3)+7$

$=x^4-37x^2-123x+110$