What is the standard form of #f(x)=(2x-3)(x-2)+(4x-5 )^2 #?
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The polynomial in standard form is #18x^2-47x+31#.
#f(x)=color(red)((2x-3)(x-2))+color(blue)((4x-5)^2)#
#color(white)(f(x))=color(red)(2x^2-4x-3x+6)+color(blue)((4x-5)(4x-5))#
#color(white)(f(x))=color(red)(2x^2-7x+6)+color(blue)(16x^2-20x-20x+25)#
#color(white)(f(x))=color(red)(2x^2-7x+6)+color(blue)(16x^2-40x+25)#
#color(white)(f(x))=color(red)(2x^2)+color(blue)(16x^2)color(red)(-7x)color(blue)(-40x)+color(red)6+color(blue)(25)#
#color(white)(f(x))=color(purple)(18x^2-47x+31)#
This is the equation of the polynomial in standard form. You can verify this by graphing the original equation and this one and seeing that they are the same parabola.
#f(x)=(2x-3)(x-2)+(4x-5)^2=color(blue)(18x^2-47x+31#
This is the standard form for a quadratic equation:
#ax^2+bx+c#.
#f(x)=(2x-3)(x-2)+(4x-5)^2#
First multiply #(2x-3)# by #(x-2)# using the FOIL method.
https://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/foil_method.html
#f(x)=2x^2-7x+6+(4x-5)^2#
Expand #(4x-5)^2# using the FOIL method.
#f(x)=2x^2-7x+6+16x^2-40x+25#
Collect like terms.
#f(x)=(2x^2+16x^2)+(-7x-40x)+(6+25)#
Combine like terms.
#f(x)=18x^2-47x+31# is in standard form for a quadratic equation:
#ax^2+bx+c#,
where:
#a=18#, #b=-47#, #c=31#
