Question

9.2 multiplying and dividing rational expressions
name ____________
date _____ period __
essential question: how can you multiply and divide rational expressions?
homework
day 1

1. explain how to multiply the rational expressions.

$\frac{x - 3}{2} cdot \frac{x^2 - 3x + 4}{x^2 - 2x}$
questioning strategies
why should you factor the numerators and the denominators before you multiply? it makes it easier to multiply because you can divide out common factors from a numerator and a denominator before multiplying.

find the products and any excluded values.

1. $\frac{x}{3x - 6} cdot \frac{x - 2}{x + 9}$
2. $\frac{5x^2 + 25x}{2} cdot \frac{4x}{x + 5}$
3. $\frac{x^2 - 2x - 15}{10x + 30} cdot \frac{3}{x^2 - 3x - 10}$
4. $\frac{x^2 - 1}{x^2 + 5x + 4} cdot \frac{x^2}{x^2 - x}$
5. $\frac{x^2 + 14x + 33}{4x} cdot \frac{x^2 - 3x}{x + 3} cdot \frac{8x - 56}{x^2 + 4x - 77}$
6. $\frac{9x^2}{x - 6} cdot \frac{x^2 - 36}{3x - 6} cdot \frac{3}{4x^2 + 24x}$

Explanation:

---

Problem 1: Explain multiplication steps

Step1: Factor all expressions

$\frac{x-3}{2} \cdot \frac{x^2-3x+4}{x(x-2)}$
(Note: $x^2-3x+4$ is irreducible over integers)

Step2: Identify excluded values

Set denominators $
eq0$: $2
eq0$, $x(x-2)
eq0 \implies x
eq0,2$

Step3: Multiply numerators/denominators

$\frac{(x-3)(x^2-3x+4)}{2x(x-2)}$

Step4: State process summary

Factor first, exclude invalid $x$, then multiply.

---

Problem 2: Find product & excluded values

Step1: Factor denominators

$\frac{x}{3(x-2)} \cdot \frac{x-2}{x+9}$

Step2: Cancel common factors

Cancel $(x-2)$: $\frac{x}{3} \cdot \frac{1}{x+9}$

Step3: Multiply remaining terms

$\frac{x}{3(x+9)}$

Step4: Find excluded values

$3(x-2)
eq0, x+9
eq0 \implies x
eq2,-9$

---

Problem 3: Find product & excluded values

Step1: Factor numerators

$\frac{5x(x+5)}{2} \cdot \frac{4x}{x+5}$

Step2: Cancel common factors

Cancel $(x+5)$: $\frac{5x}{2} \cdot 4x$

Step3: Simplify the product

$\frac{5x \cdot 4x}{2} = 10x^2$

Step4: Find excluded values

$2
eq0, x+5
eq0 \implies x
eq-5$

---

Problem 4: Find product & excluded values

Step1: Factor all expressions

$\frac{(x-5)(x+3)}{10(x+3)} \cdot \frac{3}{(x-5)(x+2)}$

Step2: Cancel common factors

Cancel $(x-5),(x+3)$: $\frac{1}{10} \cdot \frac{3}{x+2}$

Step3: Multiply remaining terms

$\frac{3}{10(x+2)}$

Step4: Find excluded values

$10(x+3)
eq0, (x-5)(x+2)
eq0 \implies x
eq-3,5,-2$

---

Problem 5: Find product & excluded values

Step1: Factor all expressions

$\frac{(x-1)(x+1)}{(x+1)(x+4)} \cdot \frac{x^2}{x(x-1)}$

Step2: Cancel common factors

Cancel $(x-1),(x+1),x$: $\frac{1}{x+4} \cdot \frac{x}{1}$

Step3: Multiply remaining terms

$\frac{x}{x+4}$

Step4: Find excluded values

$(x+1)(x+4)
eq0, x(x-1)
eq0 \implies x
eq-1,-4,0,1$

---

Problem 6: Find product & excluded values

Step1: Factor all expressions

$\frac{(x+3)(x+11)}{4x} \cdot \frac{x(x-3)}{x+3} \cdot \frac{8(x-7)}{(x+11)(x-7)}$

Step2: Cancel common factors

Cancel $(x+3),(x+11),x,(x-7)$: $\frac{1}{4} \cdot (x-3) \cdot 8$

Step3: Simplify the product

$\frac{8(x-3)}{4}=2(x-3)=2x-6$

Step4: Find excluded values

$4x
eq0, x+3
eq0, (x+11)(x-7)
eq0 \implies x
eq0,-3,-11,7$

---

Problem 7: Find product & excluded values

Step1: Factor all expressions

$\frac{9x^2}{x-6} \cdot \frac{(x-6)(x+6)}{3(x-2)} \cdot \frac{3}{4x(x+6)}$

Step2: Cancel common factors

Cancel $(x-6),(x+6),3$: $\frac{9x^2}{1} \cdot \frac{1}{x-2} \cdot \frac{1}{4x}$

Step3: Simplify the product

$\frac{9x^2}{4x(x-2)}=\frac{9x}{4(x-2)}$

Step4: Find excluded values

$x-6
eq0, 3(x-2)
eq0, 4x(x+6)
eq0 \implies x
eq6,2,0,-6$

---

Answer:

1. 1. Factor all numerators/denominators (note $x^2-3x+4$ is irreducible). 2. Identify excluded values $x

eq0,2$. 3. Multiply numerators and denominators: $\boldsymbol{\frac{(x-3)(x^2-3x+4)}{2x(x-2)}}$

1. Product: $\boldsymbol{\frac{x}{3(x+9)}}$; Excluded values: $\boldsymbol{x

eq2,-9}$

1. Product: $\boldsymbol{10x^2}$; Excluded values: $\boldsymbol{x

eq-5}$

1. Product: $\boldsymbol{\frac{3}{10(x+2)}}$; Excluded values: $\boldsymbol{x

eq-3,5,-2}$

1. Product: $\boldsymbol{\frac{x}{x+4}}$; Excluded values: $\boldsymbol{x

eq-1,-4,0,1}$

1. Product: $\boldsymbol{2x-6}$; Excluded values: $\boldsymbol{x

eq0,-3,-11,7}$

1. Product: $\boldsymbol{\frac{9x}{4(x-2)}}$; Excluded values: $\boldsymbol{x

eq6,2,0,-6}$