Consider a polynomial

$$p(x)=\left(x^{2}-1\right)\left(x^{3}-4 x^{2}+4 x\right)(x-4)\left(x^{2}-6 x+9\right)$$

such that

• $t(x)=\left(x^{3}-6 x^{2}+9 x\right)$ divides $p(x)$.
• $s(x)=\left(x^{2}-5 x+4\right)$ divides $p(x)$.
• $q(x)$ is the quotient when $p(x)$ is divided by the polynomial $z(x)=(x+1) t(x) s(x)$ Use this information to answer the given subquestions: Sub questions If $m$ is the number of distinct roots and $n$ is the number of turning points of the polynomial $p(x)$, then find the value of $m+n$ Question Ladel : iviuitiple Select Question Which of the following options is/are true? Options : $\checkmark$ The Minimum value of the quotient $q(x)$ is 0 . e $p(x)$ is an odd degree polynomial.
• End behavior $p(x) \rightarrow+\infty$ as $x \rightarrow+\infty$ , End behavior $p(x) \rightarrow+\infty$ as $x \rightarrow-\infty$CopyShow ImageTo solve the given problem, we need to analyze the polynomial $ p(x) $ and its properties. Let's start by examining the polynomial and its factors. Given: $p(x) = (x^2 - 1)(x^3 - 4x^2 + 4x)(x - 4)(x^2 - 6x + 9)$

Step 1: Factorize the polynomial

1. $ x^2 - 1 = (x - 1)(x + 1) $
2. $ x^3 - 4x^2 + 4x = x(x^2 - 4x + 4) = x(x - 2)^2 $
3. $ x - 4 $ is already a linear factor.
4. $ x^2 - 6x + 9 = (x - 3)^2 $ So, we can rewrite $ p(x) $ as: $p(x) = (x - 1)(x + 1)x(x - 2)^2(x - 4)(x - 3)^2$

Step 2: Identify the roots and turning points

Roots:

The roots of $ p(x) $ are the values of $ x $ that make $ p(x) = 0 $:

• $ x = -1 $
• $ x = 0 $
• $ x = 1 $
• $ x = 2 $ (with multiplicity 2)
• $ x = 3 $ (with multiplicity 2)
• $ x = 4 $ Thus, the distinct roots are $ -1, 0, 1, 2, 3, 4 $. Therefore, the number of distinct roots $ m $ is: $m = 6$

Turning Points:

The turning points of a polynomial are given by the critical points where the first derivative changes sign. For a polynomial of degree $ n $, the maximum number of turning points is $ n - 1 $. The degree of $ p(x) $ is: $2 + 3 + 1 + 2 = 8$ Thus, the maximum number of turning points $ n $ is: $n = 8 - 1 = 7$

Step 3: Calculate $ m + n $

$m + n = 6 + 7 = 13$

Step 4: Analyze the given options

1. The Minimum value of the quotient $ q(x) $ is 0.
   • This is true because the quotient $ q(x) $ can be zero if the remainder is zero when $ p(x) $ is divided by $ z(x) $.
2. $ p(x) $ is an odd degree polynomial.
   • This is false because $ p(x) $ is a polynomial of degree 8, which is even.
3. End behavior $ p(x) \rightarrow +\infty $ as $ x \rightarrow +\infty $.
   • This is true because the leading term $ x^8 $ dominates and $ x^8 \rightarrow +\infty $ as $ x \rightarrow +\infty $.
4. End behavior $ p(x) \rightarrow +\infty $ as $ x \rightarrow -\infty $.
   • This is true because the leading term $ x^8 $ dominates and $ x^8 \rightarrow +\infty $ as $ x \rightarrow -\infty $.

Conclusion

The value of $ m + n $ is $ 13 $. The true options are:

• The Minimum value of the quotient $ q(x) $ is 0.
• End behavior $ p(x) \rightarrow +\infty $ as $ x \rightarrow +\infty $.
• End behavior $ p(x) \rightarrow +\infty $ as $ x \rightarrow -\infty $.