I have a homework assignment that I do not know how to solve. I don't understand how to calculate $f(x)$ in this assignment.
$f(t)$ is the difference between the polynomials $2t^3-7t^2-4$ and $t^2-7t-3$.
Calculate $f(3)$.
What should I do to calculate $f(t)$?
Thanks!
$2t^3-7t^2-4-(t^2-7t-3)$
$2t^3-7t^2-4-t^2+7t+3$
$f(t)=2t^3-8t^2+7t-1$
$f(3)=2(3)^3-8(3)^2+7(3)-1$
$f(3)=54-72+21-1$
$f(3)=2$
To calculate the difference between two polynomials you have to calculate the difference of the coefficients of the similar terms. $$f(t)=(2t^3-7t^2-4)-(t^2-7t-3)= 2t^3-7t^2-4-t^2+7t+3= \\ 2t^3-(7+1)t^2+7t-(4+3) \\ \Rightarrow f(t)=2t^3-8t^2+7t-1$$
$$f(3)=2\cdot 3^3-8 \cdot 3^2+7 \cdot 3-1= \dots$$
Should be $f(t)$ not $f(x)$. Subtract the two equations to find $f(t)$. This will leave you with $2t^{3}-8t^{2}+7t-1$. Now plug in $t=3$ and you are done.
If difference isn't explicitly defined then you can follow this definition.
If $h(x)$ is difference of $f(x)$ and $g(x)$, then $h(x)= | f(x) - g(x) |$.
Modulus is taken whenever the difference function isn't stated explicitly.
This might be a tad too complex for the OP, so I suggest you read this answer lightly.
Let $F$ be a field, and let $F[x_i]$ be the vector space of polynomials of $i$ variables with coefficients in $F$. This is a vector space, and hence, for two polynomials $P(x_i),Q(x_i)\in F[x_i]$, there exists a $-Q(x_i)$ satisfying $Q(x_i)+(-Q(x_i))=0$, where $0$ is the zero polynomial. Then, the difference $\alpha(x_i)$ between the two polynomials is given by $\alpha(x_i)=P(x_i)+(-Q(x_i))$. Your scenario is the case $i=1$ and $F=\mathbf{R}$, the real numbers.
