elen fills fruit bowis at a restaurant. She puts \( \frac{2}{3} \) pint of fruit in each bowl, How many bowis can Helen serve with 29

Factor the binomial completely. \[ x^{7}-x^{3} \]

Let \( a(x)=-5 x^{3}-x^{2}+3 \), and \( b(x)=x^{2}+4 \). When dividing \( a \) by \( b \), we can find the unique quotient polynomial \( q \) and remainder polynomial \( r \) that satisfy the following equation: \[ \frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)} \] where the degree of \( r(x) \) is less than the degree of \( b(x) \). What is the quotient, \( q(x) \) ? \( q(x)=\square \) What is the remainder, \( r(x) \) ? \( r(x)=\square \)

Simplify the expressions. Enter your answers in exponential form. a. \( \frac{(-3 z)^{7}}{(-3 z)^{9}}=\square \) b. \( \frac{-3}{(-3)^{2}}=\square \) c. \( \frac{(\sqrt{5} t)^{8}}{(\sqrt{5} t)^{10}}=\square \)

Factor the binomial completely. \[ 125 r^{2}-20 \]

29. Let \[ A=\left[\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 1\end{array}\right] \] We wish to determine all \( 4 \times 3 \) matrices \( X \) for which \( A X A=A \) Observe that \[ P A Q=\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right] \] where \( P \) and \( Q \) are the invertible matrices \[ P=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 2 & 1\end{array}\right] \text { and } Q=\left[\begin{array}{rrrr}1 & -1 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1\end{array}\right] \] (a) Explain how you can use this to convert the problem of finding all \( X \) so that \( A X A=A \) to finding all \( 4 \times 3 \) matrices \( Y \) so that \( B Y B=B \), where \( B=P A Q \). Make sure to indicate how \( X \) and \( Y \) are related. (b) Find all \( Y \) so that \( B Y B=B \). (c) Use your answer in (b) to find all \( X \) so that \( A X A=A \). (d) Give 2 explicit \( X \) with \( A X A=A \).