\begin{align*}
\frac{2}{3} \cdot \frac{5}{8} = \frac{2 \cdot 5}{3 \cdot 8} \amp = \frac{10}{24}\\
\amp = \frac{5}{12} && \text{reducing the fraction to lowest terms}
\end{align*}
Therefore, \(\frac{2}{3} \cdot \frac{5}{8} = \frac{5}{12}\text{.}\)
Section 7.3 Multiplying and Dividing Rational Expressions
Recall that to multiply fractions, we simply multiply their numerators together and their denominators together (separately). Also, after doing the multiplication, we can sometimes cancel common factors to reduce the fraction to lowest terms.
Example 7.3.1. Multiplying Fractions Example.
In general, the rule for multiplying fractions is,
\begin{gather*}
\boxed{\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}}
\end{gather*}
Alternatively, we can cancel common factors before multiplying.
Example 7.3.2. Cancelling Before Multiplying.
For example, to multiply \(\frac{2}{3} \cdot \frac{5}{8}\text{,}\) we can cancel the \(2\) on the top with the \(8\) on the bottom, to make \(1\) and \(4\text{,}\)
\begin{align*}
\frac{\cancel{2}^1}{3} \cdot \frac{5}{\cancel{8}^4} \amp = \frac{1}{3} \cdot \frac{5}{4}\\
\amp = \frac{5}{12}
\end{align*}
Rational expressions are multiplied in a similar way, by multiplying their numerators together and their denominators together.
Subsection 7.3.1 Multiplying Rational Expressions
Theorem 7.3.3.
To multiply two rational expressions, multiply their numerators and multiply their denominators. That is, if \(A, B, C, D\) are polynomials, then,
\begin{gather*}
\boxed{\frac{A}{B} \cdot \frac{C}{D} = \frac{AC}{BD}}
\end{gather*}
For example,
\begin{gather*}
\frac{x + 2}{x - 5} \cdot \frac{x^2 - 1}{x^2 + 1} = \frac{(x + 2)(x^2 - 1)}{(x - 5)(x^2 + 1)}
\end{gather*}
However, if we first multiply and expand the expressions in the numerator and denominator, then the polynomials will get complicated and it will be more difficult to cancel common factors, in order to simplify. Instead, we want to factor each expression first, cancel common factors, and then multiply the result left over.
-
Factors in any numerator can be cancelled with factors in any denominator. In other words, we can cancel “up and down” but also “diagonally”.
-
Cancelling common factors result in 1.
-
A good strategy is to first focus on factoring (factor everything), then look for cancellations after.
-
Note: Multiplication can be denoted by \(\cdot\) or \(\times\text{,}\) and also sometimes by brackets. In other words, each of these means the same thing:\begin{gather*} \frac{A}{B} \cdot \frac{C}{D} \qquad \frac{A}{B} \times \frac{C}{D} \qquad \brac{\frac{A}{B}} \brac{\frac{C}{D}} \end{gather*}
Subsection 7.3.2 Dividing Rational Expressions
Recall that when dividing rational numbers, this is equivalent to multiplying by the reciprocal of the second number, and then multiply like before. You may remember this rule as “keep, change, flip” (meaning, keep the first fraction the same, change the sign to multiplication, and flip the 2nd fraction).
Example 7.3.4. Dividing Fractions Example.
For example, to divide \(\frac{3}{10} \div \frac{9}{5}\text{,}\)
\begin{align*}
\frac{3}{10} \div \frac{9}{5} \amp = \frac{3}{10} \cdot \frac{5}{9}\\
\amp = \frac{3 \cdot 5}{10 \cdot 9}\\
\amp = \frac{15}{90}\\
\amp = \frac{1}{6}
\end{align*}
Therefore, \(\frac{3}{10} \div \frac{9}{5} = \frac{1}{6}\text{.}\)
In general,
\begin{gather*}
\boxed{\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}}
\end{gather*}
Rational expressions are divided similarly, i.e. when dividing by a rational expression, multiply by its reciprocal.
Theorem 7.3.5.
To divide two rational expressions, multiply by the reciprocal of the denominator. That is, if \(A, B, C, D\) are polynomials, then,
\begin{gather*}
\boxed{\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}}
\end{gather*}
where \(B, C, D\) are non-zero.
Remember: keep change flip.
Note that in addition to \(B\) and \(D\) having to be non-zero, \(C\) also has to be non-zero, otherwise we are dividing \(\frac{A}{B}\) by 0.
Subsection 7.3.3 Multiplying and Dividing with Monomials
Exercise Group 7.3.1. Multiplying and Dividing with Monomials 1.
Multiply or divide, and simplify each expression.
(a)
\(\dfrac{3x^2}{2x}\cdot\dfrac{x^2}{12}\)Answer.
\(\dfrac{x^3}{8}\)
(b)
\(\dfrac{14}{x^2}\cdot\dfrac{x^5}{2x}\)Answer.
\(7x^2\)
(c)
\(\dfrac{3}{x}\div\dfrac{12}{x^2}\)Answer.
\(\dfrac{x}{4}\)
(d)
\(\dfrac{-4x}{9}\cdot 6\)Answer.
\(-\dfrac{8x}{3}\)
(e)
\(\dfrac{y^2}{3}\cdot\dfrac{8}{y}\)Answer.
\(\dfrac{8y}{3}\)
(f)
\(\dfrac{2x^2}{7}\cdot\dfrac{21}{x}\)Answer.
\(6x\)
(g)
\(\dfrac{x^3}{6}\div\dfrac{x^2}{-3}\)Answer.
\(-\dfrac{x}{2}\)
(h)
\(\dfrac{15x^2}{18}\cdot\dfrac{9}{5x}\)Answer.
\(\dfrac{3x}{2}\)
(i)
\(\dfrac{3x^2}{2y}\cdot\dfrac{4y^2}{9}\)Answer.
\(\dfrac{2x^2y}{3}\)
(j)
\(\dfrac{6ab}{a^2b^2}\cdot\dfrac{a^2}{b^2}\)Answer.
\(\dfrac{6a}{b^3}\)
(k)
\(\dfrac{6x^2 y}{5y^3}\cdot\dfrac{xy}{8}\)Answer.
\(\dfrac{3x^3}{20y}\)
(l)
\(\dfrac{2x^3y}{3xy^2}\cdot\dfrac{9x}{4x^2y}\)Answer.
\(\dfrac{3x}{2y^2}\)
(m)
\(\dfrac{2x}{3}\div\dfrac{x^2}{5}\)Answer.
\(\dfrac{10}{3x}\)
(n)
\(\dfrac{3x^3}{2y}\cdot\dfrac{8y^2}{9x}\)Answer.
\(\dfrac{4x^2y}{3}\)
(o)
\(\dfrac{7}{2x^3}\cdot\dfrac{-x^4}{14}\)Answer.
\(-\dfrac{x}{4}\)
(p)
\(\dfrac{-5x^2}{12}\cdot\dfrac{4}{-15x^5}\)Answer.
\(\dfrac{1}{9x^3}\)
(q)
\(\dfrac{18x^2}{10y^2}\cdot\dfrac{15y^3}{24x}\)Answer.
\(\dfrac{9xy}{8}\)
(r)
\(\dfrac{15a^2b}{4c}\cdot\dfrac{8abc}{-3}\)Answer.
\(-10a^3b^2\)
(s)
\(\dfrac{24mn^2}{8m^4n^3}\cdot\dfrac{12m^3n^2}{36m^2n}\)Answer.
\(\dfrac{1}{m^2}\)
(t)
\(\dfrac{8m^3}{3n^2}\cdot\dfrac{5m^2}{6n}\)Answer.
\(\dfrac{20m^5}{9n^3}\)
Exercise Group 7.3.2. Multiplying and Dividing with Monomials 2.
Multiply or divide, and simplify each expression.
(a)
\(\dfrac{-8x^4}{3}\div\dfrac{-6x^2}{5}\)Answer.
\(\dfrac{20x^2}{9}\)
(b)
\(\dfrac{20}{3x^5}\div\dfrac{-15}{8x^2}\)Answer.
\(-\dfrac{32}{9x^3}\)
(c)
\(\dfrac{4x^3}{-3}\div 2x^4\)Answer.
\(-\dfrac{2}{3x}\)
(d)
\(\dfrac{7a}{3}\div\dfrac{14a^2}{5}\)Answer.
\(\dfrac{5}{6a}\)
(e)
\(\dfrac{-15}{2x^2}\div\dfrac{10}{3x^4}\)Answer.
\(-\dfrac{9x^2}{4}\)
(f)
\(\dfrac{12m^2 f}{5cf}\cdot\dfrac{15c}{4m}\)Answer.
\(9m\)
(g)
\(\dfrac{-4a}{7b^3}\cdot\dfrac{-8a^4}{7}\)Answer.
\(\dfrac{32a^5}{49b^3}\)
(h)
\(\dfrac{4t^2}{3s}\div\dfrac{2t}{s^2}\)Answer.
\(\dfrac{2st}{3}\)
(i)
\(\dfrac{8m^3}{3n^2}\cdot\dfrac{5m^2}{6n}\)Answer.
\(\dfrac{20m^5}{9n^3}\)
(j)
\(\dfrac{12m}{-5t}\cdot\dfrac{8m^2}{-15}\)Answer.
\(\dfrac{32m^3}{25t}\)
(k)
\(6x^3y^4\cdot\dfrac{2xy}{-3}\)Answer.
\(-4x^4y^5\)
(l)
\(\dfrac{16ab}{9x^4y^2}\cdot\dfrac{3x^5y^4}{8a^2b^2}\)Answer.
\(\dfrac{2xy^2}{3ab}\)
(m)
\(\dfrac{6x^2y}{5mn^3}\div\dfrac{9xy}{10mn^4}\)Answer.
\(\dfrac{4xn}{3}\)
(n)
\(\dfrac{21xy}{4t^2}\cdot\dfrac{12}{7x^2y}\)Answer.
\(\dfrac{9}{xt^2}\)
(o)
\(\dfrac{6x^2 y^4}{35a^2 b^4}\div\dfrac{12x^3 y^3}{7a^4 b^4}\)Answer.
\(\dfrac{a^2 y}{10x}\)
(p)
\(\dfrac{5xy}{6x^2y}\div\dfrac{10xy}{9x^3y^2}\)Answer.
\(\dfrac{3xy}{4}\)
(q)
\(-12a^2b\div\dfrac{4ab^2}{-3ab^3}\)Answer.
\(9a^2b^2\)
(r)
\(\dfrac{3a^2 b^3}{2ab^2}\div\dfrac{9a^2 b}{14a^2}\)Answer.
\(\dfrac{7a}{3}\)
(s)
\(\dfrac{27a^9 b^3}{16xy^5}\cdot\dfrac{20x^2 y^6}{9a^9 b}\)Answer.
\(\dfrac{15b^2xy}{4}\)
(t)
\(\dfrac{2a^3 b}{c^4}\cdot\dfrac{c^2}{4a^2 b^2}\cdot\dfrac{6a^2 c^2}{a^3 c^2}\)Answer.
\(\dfrac{3}{bc^2}\)
Exercise Group 7.3.3. Multiplying and Dividing with Monomials 3.
Multiply or divide, and simplify each expression.
(a)
\(\dfrac{2(x + 1)}{3} \cdot \dfrac{x - 1}{6(x + 1)}\)Answer.
\(\dfrac{x - 1}{9}\)
(b)
\(\dfrac{(x + 1)(x - 5)}{x + 4} \cdot \dfrac{x + 4}{2(x - 5)}\)Answer.
\(\dfrac{x + 1}{2}\)
(c)
\(\dfrac{3}{x-4}\times \dfrac{x-4}{6}\)Answer.
\(\dfrac{1}{2}\)
(d)
\(\dfrac{m+2}{5}\div\dfrac{y+1}{10}\)Answer.
\(\dfrac{2(m+2)}{y+1}\)
(e)
\(\dfrac{5(x-2)}{x+1}\times \dfrac{x+1}{10}\)Answer.
\(\dfrac{x-2}{2}\)
(f)
\(\dfrac{2(x+1)}{x-2}\div \dfrac{x+1}{x-2}\)Answer.
\(2\)
(g)
\(\dfrac{4a^2b}{3(a+b)}\div\dfrac{-8ab^2}{a+b}\)Answer.
\(-\dfrac{a}{6b}\)
(h)
\(\dfrac{3(x+4)}{5x}\times\dfrac{6x^3}{2(x+4)}\)Answer.
\(\dfrac{9x^2}{5}\)
(i)
\(\dfrac{8(x-2)}{y}\cdot\dfrac{3y}{6(x-2)^2}\)Answer.
\(\dfrac{4}{x-2}\)
(j)
\(\dfrac{3x(x - 6)}{(x + 2)(x - 7)} \div \dfrac{x - 6}{x + 2}\)Answer.
\(\dfrac{3x}{x - 7}\)
(k)
\(\dfrac{x(x + 4)}{(x - 3)(x - 6)} \cdot \dfrac{(x - 6)(x + 9)}{x^2(x + 4)}\)Answer.
\(\dfrac{x + 9}{x(x - 3)}\)
(l)
\(\dfrac{3(a-b)}{(a-1)(a+5)} \cdot \dfrac{(a-5)(a+5)}{15(a-b)}\)Answer.
\(\dfrac{a-5}{5(a-1)}\)
(m)
\(\dfrac{(x-7)(x+3)}{(2x-3)(2x+3)} \cdot \dfrac{4(2x+3)}{(x+3)(x-1)}\)Answer.
\(\dfrac{4(x-7)}{(2x-3)(x-1)}\)
(n)
\(\dfrac{(x + 1)(x - 3)}{(x + 2)^2} \cdot \dfrac{2(x + 2)}{(x - 3)(x + 3)}\)Answer.
\(\dfrac{2(x + 1)}{(x + 2)(x + 3)}\)
Subsection 7.3.4 Multiplying and Dividing with Polynomials by Factoring
Exercise Group 7.3.4. Multiplying and Dividing with Polynomials by Factoring 1.
Multiply or divide, and simplify each expression.
(a)
\(\dfrac{4x+4}{3x-3}\cdot\dfrac{6x-6}{5x+5}\)Answer.
\(\dfrac{8}{5}\)
(b)
\(\dfrac{6x^3}{x+3}\cdot\dfrac{5x+15}{8x^3}\)Answer.
\(\dfrac{15}{4}\)
(c)
\(\dfrac{3x+6}{9x^2}\div\dfrac{x+2}{-3x}\)Answer.
\(-\dfrac{1}{x}\)
(d)
\(\dfrac{4x-6}{8x^2y}\cdot\dfrac{4xy}{6x-9}\)Answer.
\(\dfrac{1}{3x}\)
(e)
\(\dfrac{2x-12}{7x}\cdot\dfrac{14}{3x-18}\)Answer.
\(\dfrac{4}{3x}\)
(f)
\(\dfrac{x+2}{x^2-3x}\cdot\dfrac{8x^2}{x+2}\)Answer.
\(\dfrac{8x}{x-3}\)
(g)
\(\dfrac{4x+8}{x^2-2x}\cdot\dfrac{x-2}{x+2}\)Answer.
\(\dfrac{4}{x}\)
(h)
\(\dfrac{2x-8}{x+2}\cdot\dfrac{3x+6}{x-4}\)Answer.
\(6\)
(i)
\(\dfrac{2x-6}{x+3} \cdot \dfrac{x+3}{2}\)Answer.
\(x-3\)
(j)
\(\dfrac{7x-1}{3x} \cdot \dfrac{1}{1-7x}\)Answer.
\(-\dfrac{1}{3x}\)
(k)
\(\dfrac{x-5}{x+3} \div \dfrac{x+1}{x-2}\)Answer.
\(\dfrac{(x-5)(x-2)}{(x+3)(x+1)}\)
(l)
\(\dfrac{x^2}{2x + 1} \cdot \dfrac{6x + 3}{5x}\)Answer.
\(\dfrac{3x}{5}\)
(m)
\(\dfrac{x - 7}{10} \div \dfrac{2x - 14}{25}\)Answer.
\(\dfrac{5}{4}\)
(n)
\(\dfrac{3a - 6}{a + 2} \div \dfrac{a - 2}{a + 2}\)Answer.
\(3\)
(o)
\(\dfrac{2(x - 2)}{9x^3} \cdot \dfrac{12x^4}{2 - x}\)Answer.
\(-\dfrac{8x}{3}\)
(p)
\(\dfrac{3(x+4)^2}{2x+1} \div \dfrac{5(x + 4)}{7x + 14}\)Answer.
\(\dfrac{21(x + 4)(x + 2)}{5(2x + 1)}\)
(q)
\(\dfrac{4x - 14}{6x^2 - 5x} \cdot \dfrac{24x^2 - 20x}{16x - 56}\)Answer.
\(1\)
(r)
\(\dfrac{4x - 3}{2x^2 - 15x} \div \dfrac{8x^2 - 6x}{6x^3 - 45x^2}\)Answer.
\(\dfrac{3}{2}\)
Exercise Group 7.3.5. Multiplying and Dividing with Polynomials by Factoring 2.
Multiply or divide, and simplify each expression.
(a)
\(\dfrac{x^2+5x+6}{x^2-6x+5}\cdot\dfrac{x^2+x-30}{x^2+9x+18}\)Answer.
\(\dfrac{x+2}{x-1}\)
(b)
\(\dfrac{2x^2-8}{6x+3}\div\dfrac{6x-12}{18x+9}\)Answer.
\(x+2\)
(c)
\(\dfrac{x^2-100}{144} \cdot \dfrac{36}{x+10}\)Answer.
\(\dfrac{x-10}{4}\)
(d)
\(\dfrac{x^2+7x+12}{x^2+4x+4}\cdot\dfrac{x^2-x-6}{x^2-9}\)Answer.
\(\dfrac{x+4}{x+2}\)
(e)
\(\dfrac{x^2-3x-4}{x^2+5x} \div \dfrac{x^2-7x+12}{x^2+2x-15}\)Answer.
\(\dfrac{x+1}{x}\)
(f)
\(\dfrac{2x+10}{x^2 - 4x + 4} \div \dfrac{x^2 - 25}{x - 2}\)Answer.
\(\dfrac{2}{(x - 2)(x - 5)}\)
(g)
\(\dfrac{x^2-1}{2x-14}\cdot\dfrac{x^2-4x-21}{x^2+2x-3}\)Answer.
\(\dfrac{x+1}{2}\)
(h)
\(\dfrac{x^2+2x+1}{x-1}\cdot\dfrac{3x-3}{x+1}\)Answer.
\(3(x+1)\)
(i)
\(\dfrac{x^2+10x+16}{5x-10}\cdot\dfrac{x-2}{x^2+9x+8}\)Answer.
\(\dfrac{x+2}{5(x+1)}\)
(j)
\(\dfrac{x^2-16}{x^2-4}\cdot\dfrac{x+2}{x-4}\)Answer.
\(\dfrac{x+4}{x-2}\)
(k)
\(\dfrac{x^2-49}{x^2+5x}\cdot\dfrac{x+5}{x+7}\)Answer.
\(\dfrac{x-7}{x}\)
(l)
\(\dfrac{x^2-4}{5x+10}\cdot\dfrac{x+2}{x-2}\)Answer.
\(\dfrac{x+2}{5}\)
(m)
\(\dfrac{x^2-36}{x^2-25}\cdot\dfrac{x+5}{x-6}\)Answer.
\(\dfrac{x+6}{x-5}\)
(n)
\(\dfrac{x^2-7x}{x^2-49} \div \dfrac{3x^2}{x+7}\)Answer.
\(\dfrac{1}{3x}\)
(o)
\(\dfrac{5}{x+1} \div \dfrac{10}{x^2-1} \div (x-1)\)Answer.
\(\dfrac{1}{2}\)
(p)
\(\dfrac{x^2}{x^2-9} \div \dfrac{x}{x-3}\)Answer.
\(\dfrac{x}{x+3}\)
(q)
\(\dfrac{x^2-1}{2x-2}\cdot\dfrac{4x}{x+1}\)Answer.
\(2x\)
(r)
\(\dfrac{x^2 - 1}{x - 2} \div \dfrac{x + 1}{12 - 6x}\)Answer.
\(-6(x - 1)\)
(s)
\(\dfrac{x^2-2x-24}{x^2-36}\cdot\dfrac{x^2+5x-6}{x^2+2x-8}\)Answer.
\(\dfrac{x-1}{x-2}\)
(t)
\(\dfrac{2x^2-10x}{x^2-9x+20}\cdot\dfrac{x^2-8x+16}{4x^2}\)Answer.
\(\dfrac{x-4}{2x}\)
(u)
\(\dfrac{x^2+7x+12}{x^2+8x+16}\cdot\dfrac{x+4}{x+3}\)Answer.
\(1\)
(v)
\(\dfrac{x+3}{x+1} \cdot \dfrac{x^2-1}{x^2-9}\)Answer.
\(\dfrac{x-1}{x-3}\)
(w)
\(\dfrac{x^2 - 8x + 15}{x^2 + 4x - 15} \div \dfrac{15 - 2x - x^2}{x^2 + 11x + 18}\)Answer.
\(-\dfrac{x + 2}{x + 5}\)
(x)
\(\dfrac{x-5}{x} \div \dfrac{x^2-2x-15}{x^3}\)Answer.
\(\dfrac{x^2}{x+3}\)
(y)
\(\dfrac{8x+8}{x^2-2x+1}\cdot\dfrac{x-1}{2x+2}\)Answer.
\(\dfrac{4}{x-1}\)
(z)
\(\dfrac{x^2-25}{x+2}\cdot\dfrac{x^2-4}{x^2-7x+10}\)Answer.
\(x+5\)
(aa)
\(\dfrac{(x+1)^2}{x^2 + 2x - 3} \cdot \dfrac{(x-1)^2}{x^2 + 4x + 3}\)Answer.
\(\dfrac{(x - 1)(x + 1)}{(x + 3)^2}\)
(ab)
\(\dfrac{x^2 - 14x + 49}{x^2 - 49} \div \dfrac{3x - 21}{x + 7}\)Answer.
\(\dfrac{1}{3}\)
(ac)
\(\dfrac{2(x^2 - 7x + 12)}{x^2 - x - 6} \div \dfrac{5(x - 4)}{x^2 - 4}\)Answer.
\(\dfrac{2(x - 2)}{5}\)
(ad)
\(\dfrac{x^2 - 1}{x - 4} \cdot \dfrac{x^2 + x - 72}{x^2 + 10x + 9}\)Answer.
\(\dfrac{(x - 1)(x - 8)}{x - 4}\)
(ae)
\(\dfrac{x^2-25}{x^2-4}\cdot\dfrac{4x-8}{3x-15}\)Answer.
\(\dfrac{4(x+5)}{3(x+2)}\)
(af)
\(\dfrac{x^2 + 7x + 10}{x^2 + x - 6} \cdot \dfrac{x + 3}{x + 5}\)Answer.
\(\dfrac{x + 2}{x - 2}\)
(ag)
\(\dfrac{x^2 - 6x + 8}{x^2 - 14x + 48} \cdot \dfrac{x^2 + x - 42}{28 - 3x - x^2}\)Answer.
\(-\dfrac{x - 2}{x - 8}\)
(ah)
\(\dfrac{x - 8}{x^2 + 5x + 4} \div \dfrac{x - 8}{x^2 - 8x - 9}\)Answer.
\(\dfrac{x - 9}{x + 4}\)
(ai)
\(\dfrac{x^2-3x+2}{x^2-4} \cdot \dfrac{x+3}{x^2+3x} \div \dfrac{1}{x+2}\)Answer.
\(\dfrac{x - 1}{x}\)
(aj)
\(\dfrac{x^2-4}{x+3}\div\dfrac{4x-8}{3x+9}\)Answer.
\(\dfrac{3(x+2)}{4}\)
(ak)
\(\dfrac{7x^2}{x^2-9}\cdot \dfrac{4x+12}{14x^3}\)Answer.
\(\dfrac{2}{x(x-3)}\)
(al)
\(\dfrac{x^2-25}{x^2-16} \div \dfrac{2x-10}{4x+16}\)Answer.
\(\dfrac{2(x+5)}{x-4}\)
(am)
\(\dfrac{x^2-3x}{x^2-3x-4}\cdot\dfrac{x^2-5x+4}{x^2-2x-3}\)Answer.
\(\dfrac{x(x-1)}{(x+1)^2}\)
(an)
\(\dfrac{x^2-3x}{x^2-x-6}\cdot\dfrac{x^2+x-2}{x-x^2}\)Answer.
\(-1\)
(ao)
\((x-7)\cdot\dfrac{x^2-x}{x^2-8x+7}\)Answer.
\(x\)
(ap)
\(\dfrac{x-2}{x+3}\div\dfrac{x^2+x-2}{x^2-4}\)Answer.
\(\dfrac{(x-2)^2}{(x+3)(x-1)}\)
Exercise Group 7.3.6. Multiplying and Dividing with Polynomials by Factoring 3.
Multiply or divide, and simplify each expression.
(a)
\(\dfrac{2x+4}{x+4}\cdot\dfrac{5x^2+21x+4}{10x+2}\)Answer.
\(x+2\)
(b)
\(\dfrac{3x^2-7x+4}{12x^2-4x}\cdot\dfrac{3x-1}{15x^3-20x^2}\)Answer.
\(\dfrac{x-1}{20x^3}\)
(c)
\(\dfrac{2x^2 - 13x + 20}{x^2 - 16} \cdot \dfrac{2x^2 + 9x + 4}{6x^2 - 7x - 5}\)Answer.
\(\dfrac{2x - 5}{3x - 5}\)
(d)
\(\dfrac{4x^2-25}{2x^2-13x+20} \cdot \dfrac{x-4}{4x+10}\)Answer.
\(\dfrac{1}{2}\)
(e)
\(\dfrac{x^2+9x+8}{4x^2-9}\cdot\dfrac{2x-3}{x+1}\)Answer.
\(\dfrac{x+8}{2x+3}\)
(f)
\(\dfrac{2x^2+5x-3}{2x-3} \cdot \dfrac{x^2-1}{6x-3} \cdot \dfrac{2x-3}{x^2+2x-3}\)Answer.
\(\dfrac{x+1}{3}\)
(g)
\(\dfrac{2x^2-x-6}{3x+6} \div \dfrac{2x+3}{x+2}\)Answer.
\(\dfrac{x-2}{3}\)
(h)
\(\dfrac{9x^2-1}{x+5} \div \dfrac{3x^2-5x-2}{2-x}\)Answer.
\(\dfrac{1-3x}{x+5}\)
(i)
\(\dfrac{8x^2-2x-3}{x^2-1} \div \dfrac{2x^2-3x-2}{2x-2} \div \dfrac{3-4x}{x+1}\)Answer.
\(-\dfrac{2}{x-2}\)
(j)
\(\dfrac{x^2+6x+8}{2x^2+9x+4}\cdot\dfrac{2x^2-x-1}{x^2-3x+2}\)Answer.
\(\dfrac{x+2}{x-2}\)
(k)
\(\dfrac{14 + 17x - 6x^2}{3x^2 + 8x + 4} \div \dfrac{4x^2 - 49}{2x^2 + 11x + 14}\)Answer.
\(-1\)
(l)
\(\dfrac{6x^2+6x}{x^2-3x-4}\cdot\dfrac{2x^2-7x-4}{8x^3+4x^2}\)Answer.
\(\dfrac{3}{2x}\)
(m)
\(\dfrac{2x^2+3x-2}{2x-1}\div(4-x^2)\)Answer.
\(\dfrac{1}{2-x}\)
(n)
\(\dfrac{2x-3}{10x^2-17x+3}\cdot\dfrac{15x^3-3x^2}{6x^2-6x}\)Answer.
\(\dfrac{x}{2(x-1)}\)
(o)
\(\dfrac{2x^2 - x - 1}{x^2 - x - 6} \cdot \dfrac{6x^2 - 5x + 1}{8x^2 + 14x + 5}\)Answer.
\(\dfrac{(x - 1)(3x - 1)(2x - 1)}{(x - 3)(x + 2)(4x + 5)}\)
(p)
\(\dfrac{9y^2 - 4}{4y - 12} \div \dfrac{9y^2 + 12y + 4}{18 - 6y}\)Answer.
\(-\dfrac{3(3y - 2)}{2(3y + 2)}\)
(q)
\(\dfrac{x^2-x-12}{(x-1)^2} \cdot \dfrac{x^2-1}{2x^2+7x+3}\)Answer.
\(\dfrac{(x-4)(x+1)}{(x-1)(2x+1)}\)
(r)
\(\dfrac{x^2 - 2x}{2x^2 + x - 15} \cdot \dfrac{2x^2 - x - 10}{x^2 - 4}\)Answer.
\(\dfrac{x}{x + 3}\)
(s)
\(\dfrac{8x^2 + 8x}{16x + 16x^2 + 4x^3} \div \dfrac{x^2 - 1}{1 - x^3}\)Answer.
\(-\dfrac{2(x^2 + x + 1)}{(x + 2)^2}\)
(t)
\(\dfrac{3x^3+14x^2-5x}{4x^2+7x+3} \div \dfrac{6x^2-2x}{8x^2+2x-3}\)Answer.
\(\dfrac{(x+5)(2x-1)}{2(x+1)}\)
(u)
\(\dfrac{3x-2}{x^3+3x^2+2x} \div \dfrac{9x^2-4}{3x^2+8x+4} \div \dfrac{1}{x}\)Answer.
\(\dfrac{1}{x+1}\)
(v)
\(\dfrac{12x^2-19x+5}{4x^2-9}\cdot\dfrac{2x-3}{3x-1}\)Answer.
\(\dfrac{4x-5}{2x+3}\)
(w)
\(\dfrac{x^2 + x - 6}{(2x - 1)^2} \cdot \dfrac{x(2x - 1)^2}{x^2 + 2x - 3} \div \dfrac{x^2 - 4}{3x}\)Answer.
\(\dfrac{3x^2}{(x - 1)(x + 2)}\)
(x)
\(\dfrac{2x^2-5x-3}{2x^2-11x+15}\cdot\dfrac{4x^2-8x-5}{4x^2+4x+1}\)Answer.
\(1\)
(y)
\(\dfrac{12x^2-5x-2}{8x^2+2x-21}\div\dfrac{12x^2+x-6}{8x^2-2x-15}\)Answer.
\(\dfrac{(4x+1)(4x+5)}{(4x+7)(4x+3)}\)
Subsection 7.3.5 Multiplying and Dividing with Two Variables
Exercise Group 7.3.7. Multiplying and Dividing with Two Variables.
Multiply or divide, and simplify each expression.
(a)
\(\dfrac{x-y}{2x}\cdot\dfrac{x^2}{(x-y)^2}\)Answer.
\(\dfrac{x}{2(x-y)}\)
(b)
\(\dfrac{3p-3r}{10pr}\cdot\dfrac{20p^2r^2}{p^2-r^2}\)Answer.
\(\dfrac{6pr}{p+r}\)
(c)
\(\dfrac{x^4 y^4}{x^2 - 5x - 6} \cdot \dfrac{2x^2 - 15x + 18}{x^7 y^4}\)Answer.
\(\dfrac{2x - 3}{x^3(x + 1)}\)
(d)
\(\dfrac{x^2 - 5xy + 4y^2}{x^2 + 3xy - 28y^2} \cdot \dfrac{x^2 + 2xy + y^2}{x^2 - y^2}\)Answer.
\(\dfrac{x + y}{x + 7y}\)
(e)
\(\dfrac{4x^2 - 4y^2}{24x^2 y^5} \div \dfrac{x^2 + 2xy + y^2}{30xy^4}\)Answer.
\(\dfrac{5(x - y)}{xy(x + y)}\)
(f)
\(\dfrac{2a^2 - 12ab + 18b^2}{a^2 - 7ab + 10b^2} \div \dfrac{4a^2 - 12ab}{a^2 - 7ab + 10b^2}\)Answer.
\(\dfrac{a - 3b}{2a}\)
(g)
\(\dfrac{10x^2 + 3xy - y^2}{9x^2 - y^2} \div \dfrac{6x^2 + 3xy}{12x + 4y}\)Answer.
\(\dfrac{4(5x - y)}{3x(3x - y)}\)
(h)
\(\dfrac{a^2 - b^2}{(a + b)^2} \div \dfrac{a - b}{a^3 + b^3}\)Answer.
\(a^2 - ab + b^2\)
(i)
\(\dfrac{15m^2 + mn - 2n^2}{2n - 14m} \cdot \dfrac{7m^2 - 8mn + n^2}{5m^2 + 7mn + 2n^2}\)Answer.
\(-\dfrac{(3m - n)(m - n)}{2(m + n)}\)
(j)
\(\dfrac{x^3 - y^3}{2x^2 + 6xy - 8y^2} \cdot \dfrac{2x^2 + 10xy + 8y^2}{x^2 + xy + y^2}\)Answer.
\(x + y\)
(k)
\(\dfrac{x^3 + y^3}{4x^7 + 4x^6 y} \div \dfrac{2x^3 - 2x^2 y + 2xy^2}{8x^2 - 8y^2}\)Answer.
\(\dfrac{(x - y)(x + y)}{x^7}\)
(l)
\(\dfrac{x^2-xy-20y^2}{x^2-8xy+15y^2} \div \dfrac{x^2+2xy-8y^2}{x^2-xy-6y^2}\)Answer.
\(\dfrac{x+2y}{x-2y}\)
(m)
\(\dfrac{x^2+2xy-8y^2}{x^2+4xy+3y^2}\cdot\dfrac{x^2+2xy-3y^2}{x^2-3xy+2y^2}\)Answer.
\(\dfrac{x+4y}{x+y}\)
(n)
\(\dfrac{x^2-xy-6y^2}{x^2-4xy+3y^2}\cdot\dfrac{y^2-x^2}{x^2+3xy+2y^2}\)Answer.
\(-1\)
(o)
\(\dfrac{x^2+xy-30y^2}{x^2-xy-20y^2}\cdot\dfrac{x^2+xy-12y^2}{x^2-2xy-3y^2}\)Answer.
\(\dfrac{x+6y}{x+y}\)
(p)
\(\dfrac{6x^2+xy-2y^2}{4x^2-8xy+3y^2}\cdot\dfrac{x-y}{3x+2y}\cdot\dfrac{8x-12y}{2y-2x}\)Answer.
\(-2\)
(q)
\(\dfrac{4y-4x}{8y^3}\div\dfrac{x^2-y^2}{2x+2y}\)Answer.
\(-\dfrac{1}{y^3}\)
(r)
\(\dfrac{x^2-y^2}{3x^2+3xy}\div\dfrac{3x^2-2xy-y^2}{3x^2+6x}\)Answer.
\(\dfrac{x+2}{3x+y}\)
(s)
\(\dfrac{3x+4y}{x^2+4xy+4y^2}\div\dfrac{2}{x+2y}\)Answer.
\(\dfrac{3x+4y}{2(x+2y)}\)
(t)
\(\dfrac{x^2-4}{2y}\div\dfrac{2-x}{6xy}\)Answer.
\(-3x(x+2)\)
(u)
\(\dfrac{x^2+3xy}{x^2-xy-42y^2}\cdot\dfrac{x^2-10xy+21y^2}{x^2-9y^2}\)Answer.
\(\dfrac{x}{x+6y}\)
(v)
\(\dfrac{a^2+15ab+56b^2}{a^2-3ab-54b^2}\div\dfrac{a^2+6ab-16b^2}{a^2+4ab-12b^2}\)Answer.
\(\dfrac{a+7b}{a-9b}\)
(w)
\(\dfrac{9s^2+30st+25t^2}{25s^2-25st-6t^2}\cdot\dfrac{20s^2-49st+30t^2}{12s^2+5st-25t^2}\)Answer.
\(\dfrac{3s+5t}{5s+t}\)
(x)
\(\dfrac{4x^2-9y^2}{4x^2-12xy+9y^2} \div \dfrac{4x^2+12xy+9y^2}{9y^2-4x^2}\)Answer.
\(-1\)
(y)
\(\dfrac{\dfrac{m^2 - mn}{6m^2 + 11mn + 3n^2} \div \dfrac{m^2 - n^2}{2m^2 - mn - 6n^2}}{\dfrac{4m^2 - 7mn - 2n^2}{3m^2 + 7mn + 2n^2}}\)Answer.
\(\dfrac{m(m + 2n)}{(m + n)(4m + n)}\)
Exercise Group 7.3.8. Factor By Grouping.
Multiply or divide, and simplify each expression.
(a)
\(\dfrac{4a^2-ab-5b^2}{ax+by+ay+bx}\div(4a-8b)\)Answer.
\(\dfrac{4a-5b}{4(x+y)(a-2b)}\)
(b)
\(\dfrac{3x-3y-x^2+y^2}{4x^2-4xy+y^2}\)Answer.
\(-\dfrac{(x-y)(x+y-3)}{(2x-y)^2}\)
(c)
\(\dfrac{4x^2-y^2+10x-5y}{2x^2+xy+5x}\)Answer.
\(\dfrac{2x-y}{x}\)
(d)
\(\dfrac{x^2+x-y^2-y}{3x^2-3y^2}\div\dfrac{5x+5y+5}{7x^2y+7xy^2}\)Answer.
\(\dfrac{7xy}{15}\)
