4. In each of the following, find the quotient and the remainder when the firs polynomial is divided by the second: $$ f(x)=3 x^{3}+5 x^{2}-7 x-6 ; x+1 $$ b. $$ f(x)=4 x^{3}-5 x^{2}+4 x-17 ; x^{2}-1 $$ c. $$ f(x)=2 x^{4}+5 x^{2}-6 ; x^{2}-x+1 $$ d. $$ f(x)=x^{5}+3 x^{4}+2 x^{3}-x^{2}+2 x-7: x+2 $$

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To find the quotient and remainder when a polynomial is divided by another, we can use polynomial long division.

Let's go through each of the given polynomials:

a. $$ f(x) = 3x^3 + 5x^2 - 7x - 6 $$ divided by $$ x + 1 $$

To perform polynomial long division, we divide the highest degree term of the dividend by the highest degree term of the divisor.

The quotient and remainder can be found by performing the polynomial long division.

Let's calculate the quotient and remainder for each polynomial:
Divide the polynomials by following steps:
- step0: Using Long Division to Divide Polynomials:
  $$\frac{\left(3x^{3}+5x^{2}-7x-6\right)}{\left(x+1\right)}$$
- step1: Calculate:
  $$3x^{2}+2x-9+\frac{3}{x+1}$$
a. The quotient is $$3x^2 + 2x - 9$$ and the remainder is $$\frac{3}{x+1}$$.

b. $$ f(x) = 4x^3 - 5x^2 + 4x - 17 $$ divided by $$ x^2 - 1 $$

Let's calculate the quotient and remainder for this polynomial:
Divide the polynomials by following steps:
- step0: Using Long Division to Divide Polynomials:
  $$\frac{\left(4x^{3}-5x^{2}+4x-17\right)}{\left(x^{2}-1\right)}$$
- step1: Calculate:
  $$4x-5+\frac{8x-22}{x^{2}-1}$$
b. The quotient is $$4x - 5$$ and the remainder is $$\frac{8x - 22}{x^2 - 1}$$.

c. $$ f(x) = 2x^4 + 5x^2 - 6 $$ divided by $$ x^2 - x + 1 $$

Let's calculate the quotient and remainder for this polynomial:
Divide the polynomials by following steps:
- step0: Using Long Division to Divide Polynomials:
  $$\frac{\left(2x^{4}+5x^{2}-6\right)}{\left(x^{2}-x+1\right)}$$
- step1: Calculate:
  $$2x^{2}+2x+5+\frac{3x-11}{x^{2}-x+1}$$
c. The quotient is $$2x^2 + 2x + 5$$ and the remainder is $$\frac{3x - 11}{x^2 - x + 1}$$.

d. $$ f(x) = x^5 + 3x^4 + 2x^3 - x^2 + 2x - 7 $$ divided by $$ x + 2 $$

Let's calculate the quotient and remainder for this polynomial:
Divide the polynomials by following steps:
- step0: Using Long Division to Divide Polynomials:
  $$\frac{\left(x^{5}+3x^{4}+2x^{3}-x^{2}+2x-7\right)}{\left(x+2\right)}$$
- step1: Calculate:
  $$x^{4}+x^{3}-x+4+\frac{-15}{x+2}$$
d. The quotient is $$x^4 + x^3 - x + 4$$ and the remainder is $$\frac{-15}{x+2}$$.
a. Quotient: $$3x^2 + 2x - 9$$, Remainder: $$\frac{3}{x+1}$$ b. Quotient: $$4x - 5$$, Remainder: $$\frac{8x - 22}{x^2 - 1}$$ c. Quotient: $$2x^2 + 2x + 5$$, Remainder: $$\frac{3x - 11}{x^2 - x + 1}$$ d. Quotient: $$x^4 + x^3 - x + 4$$, Remainder: $$\frac{-15}{x+2}$$

4. In each of the following, find the quotient and the remainder when the firs polynomial is divided by the second: \( f(x)=3 x^{3}+5 x^{2}-7 x-6 ; x+1 \) b. \( f(x)=4 x^{3}-5 x^{2}+4 x-17 ; x^{2}-1 \) c. \( f(x)=2 x^{4}+5 x^{2}-6 ; x^{2}-x+1 \) d. \( f(x)=x^{5}+3 x^{4}+2 x^{3}-x^{2}+2 x-7: x+2 \)

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