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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

Best Answer 

 #1
avatar+1493 
+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
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1+0 Answers

 #1
avatar+1493 
+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

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