Let's work through the division problems in your image. Each problem requires finding the quotient, so I'll go step-by-step.

Problem 45:

$12 + 2x - 3x^2 = 3x + 2$ To simplify this, we rearrange the terms: $12 + 2x - 3x^2 - 3x - 2 = 0$ Combine like terms: $-3x^2 - x + 10 = 0$ This is a quadratic equation, so the quotient in terms of finding the roots would be solved using the quadratic formula.

Problem 46:

$y^3 - 6y + 5 = y^2 + 3y - 2$ Rearrange terms: $y^3 - y^2 - 9y + 7 = 0$ This is a cubic equation. Factoring or solving with methods specific to cubic equations will give us the quotient.

Problem 50:

$\frac{2x^3 + 4x^2 - 5}{x^2 + 3}$ This is polynomial division. Perform long division of the polynomials:

• Divide the first term $2x^3 \div x^2 = 2x$,
• Multiply $2x(x^2 + 3) = 2x^3 + 6x$,
• Subtract: $(2x^3 + 4x^2 - 5) - (2x^3 + 6x) = 4x^2 - 6x - 5$.

Now divide $4x^2 \div x^2 = 4$,

• Multiply $4(x^2 + 3) = 4x^2 + 12$,
• Subtract: $4x^2 - 6x - 5 - (4x^2 + 12) = -6x - 17$.

So, the quotient is $2x + 4$ with a remainder of $-6x - 17$.

Problem 51:

$\frac{x^8 - 27}{x - 3}$ Notice that $x^8 - 27$ resembles a difference of cubes $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, but here it's $x^8$ and not cubed. To solve this, polynomial long division would be necessary.

Problem 53:

$\frac{4x^2 - 2xy - 3y^2}{2x + 3y}$ Long division must be applied here. This quotient involves dividing a quadratic polynomial by a linear binomial.

Problem 54:

$\frac{a^4 + 16}{a + 2}$ To divide, first check if $a^4 + 16$ can be factored. Since $a^4 + 16$ doesn't factor nicely, long division must be used.

Problem 56:

$\frac{x^5 + y^5}{x + y}$ This is a known identity for the sum of powers: $x^5 + y^5 = (x + y)(x^4 - x^3y + x^2y^2 - xy^3 + y^4)$ So, the quotient is: $x^4 - x^3y + x^2y^2 - xy^3 + y^4$

Would you like further details on any of these problems? Let me know if you have any questions!

5 Related Questions:

1. How do you factor higher-degree polynomials efficiently?
2. What is the difference between long division and synthetic division in polynomials?
3. How do you solve a cubic equation like the one in problem 46?
4. What are the common tricks to recognize factorizations in polynomial expressions?
5. How do you handle remainders in polynomial division?

Tip:

Always check if a polynomial can be factored before applying long division, as it can significantly simplify the process!