Question

practice
example 1
simplify each expression.

1. $\frac{9y^4 + 6y^2 + 3y}{3y}$
2. $(4f^5 - 6f^4 + 12f^3 - 8f^2)(4f)^{-1}$
3. $(63k^4 - 9k^3) \div (3k)$
4. $(4a^2b^2 - 8a^3b + 3a^4) \div (2a^2)$

examples 2 and 3
simplify by using long division.

1. $(n^2 + 7n + 10) \div (n + 5)$
2. $(d^2 + 4d + 3)(d + 1)^{-1}$
3. $(2t^2 + 13t + 15) \div (t + 6)$
4. $(6y^2 + y - 2)(2y - 1)^{-1}$
5. $(4g^2 - 9) \div (2g + 3)$
6. $(2x^2 - 5x - 4) \div (x - 3)$

examples 4 and 5
simplify using synthetic division.

1. $(3v^2 - 7v - 10)(v - 4)^{-1}$
2. $(3t^3 + 4t^2 - 32t - 5t - 20)(t + 4)^{-1}$
3. $\frac{y^2 + 6}{y + 2}$
4. $\frac{2x^3 - x^2 - 18x + 32}{2x - 6}$
5. $(4a^3 - a^2 + 2a) \div (3a - 1)$
6. $(8c^3 + 6c^2 - 2c + 8)(c + 2)^{-1}$

Explanation:

Let's solve problem 1: \(\frac{9y^{3}+6y^{2}+3y}{3y}\)

Step 1: Divide each term by \(3y\)

We can split the fraction into three separate fractions: \(\frac{9y^{3}}{3y}+\frac{6y^{2}}{3y}+\frac{3y}{3y}\)

Step 2: Simplify each fraction

For \(\frac{9y^{3}}{3y}\), divide the coefficients and subtract the exponents of \(y\): \( \frac{9}{3}y^{3 - 1}=3y^{2}\)
For \(\frac{6y^{2}}{3y}\): \( \frac{6}{3}y^{2 - 1}=2y\)
For \(\frac{3y}{3y}\): \( \frac{3}{3}y^{1 - 1}=1\) (since \(y^{0} = 1\))

Step 3: Combine the simplified terms

Adding the simplified terms together, we get \(3y^{2}+2y + 1\)

Answer:

\(3y^{2}+2y + 1\)