Answer:
Let's solve the exercises 14, 16, and 17 from the image:
### Exercise 14
Perform the following operations:
\[ P(x) + Q(x) \]
\[ P(x) - Q(x) \]
\[ 4P(x) - 3R(x) \]
\[ P(x) \cdot Q(x) \]
\[ Q(x) : R(x) \]
We would need the definitions of \( P(x) \), \( Q(x) \), and \( R(x) \) to solve these operations.
### Exercise 16
Expand using notable identities:
\[ (3x^5 - 7x^3)^2 \]
Using the identity \((a - b)^2 = a^2 - 2ab + b^2\):
\[ (3x^5 - 7x^3)^2 = (3x^5)^2 - 2(3x^5)(7x^3) + (7x^3)^2 \]
\[ = 9x^{10} - 42x^8 + 49x^6 \]
### Exercise 17
Perform the following division using Ruffini's rule:
\[ 6x^6 - 5x^5 + 4x^3 - 2x^2 - 7x + 3 \div (x - 3) \]
Using Ruffini's rule, set up the coefficients of the polynomial \( 6, -5, 0, 4, -2, -7, 3 \) and divide by the root \( x = 3 \):
1. Write the coefficients: \( 6, -5, 0, 4, -2, -7, 3 \)
2. The root \( x = 3 \).
Let's perform the Ruffini's division step by step:
| 3 | 6 | -5 | 0 | 4 | -2 | -7 | 3 |
|---|---|----|---|---|----|----|---|
| | | 18 | 39|117|363 |1081|3234|
| | 6 | 13 | 39|121|359 |1074|3237|
The resulting polynomial is \( 6x^5 + 13x^4 + 39x^3 + 121x^2 + 359x + 1074 \) with a remainder of \( 3237 \).
Thus, the quotient is:
\[ 6x^5 + 13x^4 + 39x^3 + 121x^2 + 359x + 1074 \]
and the remainder is:
\[ 3237 \]
If you need more detailed steps or further assistance, please let me know!