Question

given the following function:$f(x) = 9x^3 - 5x^2 + 3x + 5$step 3 of 3 : find $f(x + c)$.answer$f(x + c) = $

Explanation:

Step1: Substitute $x+c$ into $f(x)$

$f(x+c) = 9(x+c)^3 - 5(x+c)^2 + 3(x+c) + 5$

Step2: Expand $(x+c)^3$ and $(x+c)^2$

$(x+c)^3 = x^3 + 3x^2c + 3xc^2 + c^3$, $(x+c)^2 = x^2 + 2xc + c^2$

Step3: Distribute coefficients to expansions

$9(x^3 + 3x^2c + 3xc^2 + c^3) = 9x^3 + 27x^2c + 27xc^2 + 9c^3$
$-5(x^2 + 2xc + c^2) = -5x^2 -10xc -5c^2$
$3(x+c) = 3x + 3c$

Step4: Combine all expanded terms

$f(x+c) = 9x^3 + 27x^2c + 27xc^2 + 9c^3 -5x^2 -10xc -5c^2 + 3x + 3c + 5$

Step5: Group like terms (optional, for clarity)

$f(x+c) = 9x^3 + (27c - 5)x^2 + (27c^2 -10c + 3)x + (9c^3 -5c^2 + 3c + 5)$

Answer:

$9x^3 + (27c - 5)x^2 + (27c^2 - 10c + 3)x + 9c^3 - 5c^2 + 3c + 5$