+619 If $f(x) = x^3 - 6x^2 + 3x - 4$, $g(x) = x^3 + 5x^2 + 9x - 2$, then find the constant term of $f(g(x))$.
+2446 A constant term is the term that is unchanging. 6, for example, is a constant. 12.5 is a constant; they don't change.
To figure out f(g(x)), let's break this down bit by bit. Since we already know that g(x)=x3+5x2+9x−2, this means that f(g(x))=f(x3+5x2+9x−2).
| f(x)=x3−6x2+3x−4 | Now, substitute f(g(x)) into all instances of x. |
| f(g(x))=(x3+5x2+9x−2)3−6(x3+5x2+9x−2)2+3(x3+5x2+9x−2)−4 | Luckily, however, we only care about the constant terms. Let's deal with one term at a time. |
| (x3+5x2+9x−2)3 | We only care about the constant term, so do -2^3=-8 |
| (−2)3=−8 | Let's worry about the second term. |
| −6(x3+5x2+9x−2)2 | Let's do the exact same process. |
| −6∗(−2)2=−6∗4=−24 | And of course, the next term, as well. |
| 3(x3+5x2+9x−2) | |
| 3∗−2=−6 | And the final term, which happens to be a constant. |
| −4=−4 | Now, add all of those together to get the constant term. |
| −8−24−6−4=−42 | This is value of the constant term. |
+2446 A constant term is the term that is unchanging. 6, for example, is a constant. 12.5 is a constant; they don't change.
To figure out f(g(x)), let's break this down bit by bit. Since we already know that g(x)=x3+5x2+9x−2, this means that f(g(x))=f(x3+5x2+9x−2).
| f(x)=x3−6x2+3x−4 | Now, substitute f(g(x)) into all instances of x. |
| f(g(x))=(x3+5x2+9x−2)3−6(x3+5x2+9x−2)2+3(x3+5x2+9x−2)−4 | Luckily, however, we only care about the constant terms. Let's deal with one term at a time. |
| (x3+5x2+9x−2)3 | We only care about the constant term, so do -2^3=-8 |
| (−2)3=−8 | Let's worry about the second term. |
| −6(x3+5x2+9x−2)2 | Let's do the exact same process. |
| −6∗(−2)2=−6∗4=−24 | And of course, the next term, as well. |
| 3(x3+5x2+9x−2) | |
| 3∗−2=−6 | And the final term, which happens to be a constant. |
| −4=−4 | Now, add all of those together to get the constant term. |
| −8−24−6−4=−42 | This is value of the constant term. |
