For how many integers x in the range -50 \le x \le 50 is the value of the expression x^5 -… — GMAT

For how many integers $x$ in the range $-50 \le x \le 50$ is the value of the expression $x^5 - 5x^3 + 4x + 3$ an even integer?

Cevap: 0

There are 0 integers in the specified range for which the expression evaluates to an even integer.

Factoring

x^5 - 5x^3 + 4x

yields

(x - 2)(x - 1)x(x + 1)(x + 2)

, which is the product of 5 consecutive integers. The product of consecutive integers is always even for any integer

x

. Adding 3 (an odd integer) to an even integer results in an odd integer for all values of

x

. Consequently, zero integers in the given range result in an even value.

Factor the variable portion of the polynomial

x^5 - 5x^3 + 4x + 3

The expression rewrites as

(x - 2)(x - 1)x(x + 1)(x + 2) + 3

Factoring helps identify structural properties such as consecutive terms.

Analyze the parity of the product of 5 consecutive integers

(x - 2)(x - 1)x(x + 1)(x + 2)

The product is always an even integer for all integer values of

x

Any sequence of 5 consecutive integers includes multiple even numbers. Even when

x = 0

, the product equals 0, which is an even integer.

Determine the parity of the full expression by adding the constant term 3.

Even integer + 3 (odd integer) = odd integer.

The sum of an even integer and an odd integer is always an odd integer.

Count the number of integer values of

x

-50 \le x \le 50

that yield an even integer.

0 integers.

Since the expression evaluates to an odd integer for every integer

x

, no integer value can produce an even integer.

Parity of consecutive integer products and addition rules for even and odd integers.

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