Let S be the set of all real solutions to the polynomial equation (x^2 - 5x + 6)(x^2 - 7x … — GMAT

Let $S$ be the set of all real solutions to the polynomial equation $(x^2 - 5x + 6)(x^2 - 7x + 12) = (x - 3)^2(x - 2)$ . What is the sum of all elements in set $S$ ?

A
5
B
7
C
8
10Answer
E
13

Answer

The sum of all distinct real elements in set

S

is 10.

The correct answer is obtained by completely factoring both quadratic expressions on the left side into

(x - 2)(x - 3)^2(x - 4)

. Moving all terms to one side gives

(x - 2)(x - 3)^2(x - 4) - (x - 3)^2(x - 2) = 0

. Factoring out

(x - 2)(x - 3)^2

yields

(x - 2)(x - 3)^2(x - 5) = 0

. The distinct solutions are 2, 3, and 5. Their sum is 10.

Step-by-Step Solution

Factor the quadratic expressions on the left side of the equation.

x^2 - 5x + 6 = (x - 2)(x - 3)

and

x^2 - 7x + 12 = (x - 3)(x - 4)

. Substituting these into the left side yields

(x - 2)(x - 3)^2(x - 4)

Factoring simplifies higher-degree polynomial expressions into linear factors.

Set up the equation with all terms on one side to avoid dividing by variable expressions.

(x - 2)(x - 3)^2(x - 4) - (x - 3)^2(x - 2) = 0

Dividing by variable expressions can cause the loss of valid roots.

Factor out the common polynomial factor

(x - 2)(x - 3)^2

(x - 2)(x - 3)^2 [(x - 4) - 1] = 0

, which simplifies to

(x - 2)(x - 3)^2(x - 5) = 0

Factoring allows finding all roots by setting each factor equal to zero.

Identify the distinct real solutions forming set

S

and compute their sum.

The roots are

x = 2

x = 3

, and

x = 5

. Thus,

S = \{2, 3, 5\}

. The sum of elements in

S

2 + 3 + 5 = 10

A set contains only unique elements, so repeated roots are counted only once.

Key Concept

Factoring polynomial equations completely without dividing by variable expressions

Estimated Time:2m 0s

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