Dealing with the Third Degree

Quarter Wit, Quarter WisdomOne of the basic things you need to know before you start your GMAT preparation is how to solve quadratic equations i.e. factorize the quadratic  and equate each factor to 0 to get the possible values that x can take. Today we will discuss how you can solve a third degree equation.

Say an equation such as x^3 – 6x^2 + 11x – 6 = 0.

How do we get the values of x which satisfy this equation?

If you do get a third degree equation, it will have one very easy root such as 0 or 1 or -1 or 2 or -2 etc. Try a few of these values to get the first root. Here x = 1 works. It is easy to see since you have two 6s, a 1 and an 11 as the coefficients.

Putting x = 1: (1)^3 – 6(1)^2 + 11(1) – 6 = 0

So you know that (x – 1) is a factor. Now figure out the quadratic which when multiplied by (x – 1) gives the third degree expression
(x – 1)(ax^2 + bx + c) = x^3 – 6x^2 + 11x – 6

How do we figure out the values of a, b and c? Let’s see.

Coefficient of x^3 on right hand side is 1. So you know that all you need is x^2 so that it multiplies with x to give x^3 on left hand side too. So ‘a’ must be 1.

(x – 1)(x^2 + bx + c) = x^3 – 6x^2 + 11x – 6

The constant term, ‘c’, is easy to figure out too. It should multiply with -1 to give -6 on right hand side. Hence c must be 6.

(x – 1)(x^2 + bx + 6) = x^3 – 6x^2 + 11x – 6

Getting the middle term is slightly more complicated. bx multiplies with x to give x^2 term and you also get the x^2 term by multiplying -1 with x^2. You have -6x^2 on right hand side so you need the same on the left hand side too. You already have -x^2 (by multiplying -1 with x^2) so you need another -5x^2 from bx^2. So b must be -5.

(x – 1)(x^2 – 5x + 6) = x^3 – 6x^2 + 11x – 6
Now you just factorize the quadratic in the usual way. Let’s see how exactly we would do it using a question.

Question: Is x^3 + 2x^2 – 5x – 6 < 0

Statement 1: -3 < x <= -1
Statement 2: -1 <= x < 2

Solution:

We know how to deal with inequalities with multiple factors (discussed here). But the inequality is not split into factors here so we will have to do it on our own.

Let’s first try to find the simple root that this expression must have. Try x = 1, -1 etc. We see that when we put x = -1, the expression becomes 0.

x^3 + 2x^2 – 5x – 6 = (-1)^3 + 2(-1)^2 – 5(-1) – 6 = 0

So the first factor we get is (x + 1).

(x + 1)(ax^2 + bx + c) = x^3 + 2x^2 – 5x – 6

a must be 1 since we have x^3 on the right hand side.

c must be -6 since we have -6 on the right hand side.

(x + 1)(x^2 + bx – 6) = x^3 + 2x^2 – 5x – 6

bx^2 + x^2 = 2x^2

So b must be 1.

We get: (x + 1)(x^2 + x – 6) which is equal to (x + 1)(x + 3)(x – 2) (after splitting the quadratic)

So the question becomes:

Is (x + 1)(x + 3)(x – 2) < 0?

We already know how to deal with inequalities with multiple factors. The transition points here will be -3, -1 and 2. The expression will be negative in the ranges -1 < x < 2 and x < -3

Statement 1: -3 < x <= -1

In this region, the expression is positive (when -3 < x < -1) or 0 (when x = -1). Hence it will certainly not be negative. This is sufficient to answer the question with ‘No’. Hence statement 1 is sufficient to answer the question.

Statement 2: -1 <= x < 2

In this region, the expression is negative (when -1 < x < 2) or 0 (when x = -1). We cannot say for certain whether it will be negative or not. Hence statement 2 alone is not sufficient to answer the question.

Answer (A)

It’s not hard to deal with third degree equations. All you have to do is bring it down to second degree by figuring out one root and then the problem is in a format you already know.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

2 Responses

  1. Vaishno says:

    Thank you for the great explanation. You make the difficult math look so easy.

    – + – +
    ———————-(3)————————(1)—————————2—————-

    Does the number line look similar to this one? And if does, then why is statement 2 not sufficient?

    -1 <= x < 2 gives us the fact that the equation is negative, therefore sufficient to answer the question.

    Am I missing something here? Is it because we are not including -3 here? so we can not conclude from the statement 2 that the equation will definitely be negative?

    On the contrary, in statement 1 we already know that 2 is positive and 3-<x<=-1 is positive as well. So we can answer with a definite no.

  2. Vaishno says:

    Oh! I think formatting ruined my number line. I suppose you already know where the + and – sign sit on the number line, since I learned this approach from you.

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