Answer:
Given the following:
[tex]X = r+2[/tex]
[tex]Y = 2r -9[/tex] and
[tex]Z = r^2+17r+30[/tex]
Simplify: [tex][X.Y-Z] \div X[/tex]
First simplify: [tex]X \cdot Y[/tex]
[tex]X \cdot Y = (r+2) \cdot (2r-9) = r(2r-9)+2(2r-9)[/tex]
The distributive property says that:
[tex]a \cdot (b+c) = a\cdot b+ a\cdot c[/tex]
[tex]r(2r-9)+2(2r-9) = 2r^2-9r+4r-18 = 2r^2-5r-18[/tex]
∴[tex]X \cdot Y=2r^2-5r-18[/tex]
[tex]X\cdot Y -Z = 2r^2-5r-18 -(r^2+17r+30) =2r^2-5r-18 -r^2-17r- 30[/tex]
Combine like terms;
[tex]X\cdot Y -Z =r^2-22r-48=r^2-24r+2r-48 = r(r-24)+2(r-24) = (r+2)(r-24)[/tex]
Then;
[tex]\frac{X \cdot Y -Z}{X} = \frac{ (r+2)(r-24)}{r+2} = r-24[/tex]
Therefore, the simplified form of [tex][X.Y-Z] \div X[/tex] is r- 24
