What is the number of the x-intercepts of y=x^4-3x^3+2x^2?

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Mon Feb 04, 2019 4:20 am

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[GMAT math practice question]

What is the number of the x-intercepts of y=x^4-3x^3+2x^2?

A. one
B. two
C. three
D. four
E. five

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Mon Feb 04, 2019 4:27 am

Max@Math Revolution wrote:[GMAT math practice question]

What is the number of the x-intercepts of y=x^4-3x^3+2x^2?

A. one
B. two
C. three
D. four
E. five

$$?\,\,\,:\,\,\,\# \,\,x\,\,\,{\rm{for}}\,\,\,\,{x^4} - 3{x^3} + 2{x^2} = 0\,\,\,\left( * \right)$$
$$\left( * \right)\,\,\,\,\, \Rightarrow \,\,\,0 = {x^2}\left( {{x^2} - 3x + 2} \right) = {x^2}\left( {x - 1} \right)\left( {x - 2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x = 0,1\,\,{\rm{or}}\,\,2$$

The correct answer is therefore (C).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed Feb 06, 2019 6:55 am

=>

y = x^4-3x^3+2x^2
=> y = x^2(x^2-3x+2)
=> y = x^2(x-1)(z-2)
The x-intercepts occur when y = 0. This occurs when x = 0, x = 1 and x = 2.
There are three x-intercepts.

Therefore, the answer is C.
Answer: C

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8246; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Wed Feb 06, 2019 6:22 pm

Max@Math Revolution wrote:[GMAT math practice question]

What is the number of the x-intercepts of y=x^4-3x^3+2x^2?

A. one
B. two
C. three
D. four
E. five

The number of the x-intercepts of a function is equivalent to the number of distinct zeros of the function. To determine the x-intercepts, we set the function equal to zero and solve for x:

x^4-3x^3+2x^2 = 0

x^2(x^2 - 3x + 2) = 0

x^2(x - 1)(x - 2) = 0

x = 0 or x = 1 or x = 2

We see that the function has three distinct zeros; thus, it has three x-intercepts.

Answer: C

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