What is the remainder when the function #f(x)=x^3-4x^2+12# is divided by (x+2)?

2 Answers
Jan 11, 2018

#color(blue)(-12)#

Explanation:

The Remainder theorem states that, when #f(x)# is divided by #(x-a)#

#f(x)=g(x)(x-a)+r#

Where #g(x)# is the quotient and #r# is the remainder.

If for some #x# we can make #g(x)(x-a)=0#, then we have:

#f(a)=r#

From example:

#x^3-4x^2+12=g(x)(x+2)+r#

Let #x=-2#

#:.#

#(-2)^3-4(-2)^2+12=g(x)((-2)+2)+r#

#-12=0+r#

#color(blue)(r=-12)#

This theorem is just based on what we know about numerical division. i.e.

The divisor x the quotient + the remainder = the dividend

#:.#

#6/4=1# + remainder 2.

#4xx1+2=6#

Jan 11, 2018

#"remainder "=-12#

Explanation:

#"using the "color(blue)"remainder theorem"#

#"the remainder when "f(x)" is divided by "(x-a)" is "f(a)#

#"here "(x-a)=(x-(-2))rArra=-2#

#f(-2)=(-2)^3-4(-2)^2+12=-12#