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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))\$.

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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

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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
Sort:

#1
+1094
+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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