If $f(x) = x^3 - 6x^2 + 3x - 4$, $g(x) = x^3 + 5x^2 + 9x - 2$, then find the constant term of $f(g(x))$.

michaelcai
Sep 17, 2017

#1**+1 **

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 \(g(x)=x^3+5x^2+9x-2\), this means that \(f(g(x))=f(x^3+5x^2+9x-2)\).

\(f(x)=x^3-6x^2+3x-4\) | Now, substitute f(g(x)) into all instances of x. |

\(f(g(x))=(x^3+5x^2+9x-2)^3-6(x^3+5x^2+9x-2)^2+3(x^3+5x^2+9x-2)-4\) | Luckily, however, we only care about the constant terms. Let's deal with one term at a time. |

\((x^3+5x^2+9x-2)^3\) | We only care about the constant term, so do -2^3=-8 |

\((-2)^3=-8\) | Let's worry about the second term. |

\(-6(x^3+5x^2+9x-2)^2\) | Let's do the exact same process. |

\(-6*(-2)^2=-6*4=-24\) | And of course, the next term, as well. |

\(3(x^3+5x^2+9x-2)\) | |

\(3*-2=-6\) | 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. |

TheXSquaredFactor
Sep 17, 2017

#1**+1 **

Best Answer

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 \(g(x)=x^3+5x^2+9x-2\), this means that \(f(g(x))=f(x^3+5x^2+9x-2)\).

\(f(x)=x^3-6x^2+3x-4\) | Now, substitute f(g(x)) into all instances of x. |

\(f(g(x))=(x^3+5x^2+9x-2)^3-6(x^3+5x^2+9x-2)^2+3(x^3+5x^2+9x-2)-4\) | Luckily, however, we only care about the constant terms. Let's deal with one term at a time. |

\((x^3+5x^2+9x-2)^3\) | We only care about the constant term, so do -2^3=-8 |

\((-2)^3=-8\) | Let's worry about the second term. |

\(-6(x^3+5x^2+9x-2)^2\) | Let's do the exact same process. |

\(-6*(-2)^2=-6*4=-24\) | And of course, the next term, as well. |

\(3(x^3+5x^2+9x-2)\) | |

\(3*-2=-6\) | 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. |

TheXSquaredFactor
Sep 17, 2017