Question

let $f(x)=x^2 - 2x - 15$ and $g(x)=x^2 - 7x + 10$. find $(f \cdot g)(x)$. $(f \cdot g)(x)=$

Explanation:

Step1: Define product of functions

$(f \cdot g)(x) = f(x) \cdot g(x)$

Step2: Substitute given functions

$(f \cdot g)(x) = (x^2 - 2x - 15)(x^2 - 7x + 10)$

Step3: Distribute first polynomial terms

$= x^2(x^2 - 7x + 10) - 2x(x^2 - 7x + 10) - 15(x^2 - 7x + 10)$

Step4: Expand each distributed term

$= x^4 - 7x^3 + 10x^2 - 2x^3 + 14x^2 - 20x - 15x^2 + 105x - 150$

Step5: Combine like terms

$= x^4 + (-7x^3 - 2x^3) + (10x^2 + 14x^2 - 15x^2) + (-20x + 105x) - 150$
$= x^4 - 9x^3 + 9x^2 + 85x - 150$

Answer:

$x^4 - 9x^3 + 9x^2 + 85x - 150$