To multiply the given expressions and convert them into the form [tex]\(\frac{a z + b}{c z + d}\)[/tex], follow these steps:
1. Factor the numerator and denominator of each fraction:
- For the numerator of the first fraction: [tex]\(2x^2 + 2x - 21\)[/tex]
- This factors as [tex]\((2x - 3)(x + 7)\)[/tex].
- For the denominator of the first fraction: [tex]\(-2x^2 - 2x + 12\)[/tex]
- This factors as [tex]\(-1 \cdot ((2x^2 + 2x - 12) \rightarrow (2x^2 + 6x - 4x - 12) \rightarrow 2x(x + 3) -4(x + 3) = (x + 3)(2x - 4)) = -1 \cdot (2x - 3)(x + 2)\)[/tex].
- For the numerator of the second fraction: [tex]\(2x^2 + 25x + 63\)[/tex]
- This factors as [tex]\((2x + 9)(x + 7)\)[/tex].
- For the denominator of the second fraction: [tex]\(6x^2 + 7x - 49\)[/tex]
- This factors as [tex]\((3x - 7)(2x + 7)\)[/tex].
2. Write the expression with factored forms:
[tex]\[
\left( \frac{(2x - 3)(x + 7)}{-(2x - 3)(x + 2)} \right) \cdot \left( \frac{(2x + 9)(x + 7)}{(3x - 7)(2x + 7)} \right)
\][/tex]
3. Cancel out the common factors:
- Notice [tex]\((2x - 3)\)[/tex] appears in both the numerator and denominator, and [tex]\((x + 7)\)[/tex] appears in both the numerator and denominator.
Which simplifies to:
[tex]\[
\frac{-(x + 7)(2x + 9)}{(x + 2)(3x - 7)(2x + 7)}
\][/tex]
4. From the simplified fraction, express in the form [tex]\(\frac{a z + b}{c z + d}\)[/tex]:
Since [tex]\(a=1\)[/tex]:
- Numerator: [tex]\(-(x + 7)(2x + 9) \rightarrow - (2x^2 + 23x + 63) \rightarrow - 2x^2 - 23x - 63\)[/tex]
- Coefficient [tex]\(b\)[/tex] is [tex]\(-43\)[/tex].
- Denominator stays the same, giving:
[tex]\((x + 2)(3x - 7)(2x + 7)\)[/tex]
So,
- Coefficient [tex]\(c\)[/tex] is [tex]\(3\)[/tex].
- Coefficient [tex]\(d\)[/tex] is [tex]\(-98\)[/tex].
Therefore:
[tex]$b = -43$[/tex]
[tex]$c = 3$[/tex]
[tex]$d = -98$[/tex]
Final answers:
[tex]$b = -43$[/tex]
[tex]$c = 3$[/tex]
[tex]$d = -98$[/tex]
