If the polynomial 2x^2 + kx - 12 can be factored as the product of two binomials with inte… — ACT

If the polynomial $2x^2 + kx - 12$ can be factored as the product of two binomials with integer coefficients, what is the greatest possible integer value of $k$ ?

Cevap: 23

Cevap

The greatest possible integer value of

k

is 23.

The greatest possible value of

k

is 23. This is found by setting the factored form of the expression as

(2x + a)(x + b) = 2x^2 + (2b + a)x + ab

. We find that

ab = -12

and

k = 2b + a

. The possible integer pairs for

(a, b)

that multiply to

-12

are tested to find the maximum sum of

2b + a

. The maximum is achieved when

b = 12

and

a = -1

, yielding

2(12) - 1 = 23

Adım Adım Çözüm

Set up the algebraic representation for the binomial factors with integer coefficients.

The polynomial must factor into the form

(2x + a)(x + b)

where

a

and

b

are integers.

Since the leading coefficient is 2 (a prime number), the coefficients of the linear terms in the binomial factors must be 2 and 1 to obtain a product of

2x^2

Expand the product of the binomials to relate the parameters

a

and

b

to the original expression.

(2x + a)(x + b) = 2x^2 + (2b + a)x + ab

, which implies

ab = -12

and

k = 2b + a

By equating the coefficients of corresponding terms in the expanded expression and the original polynomial, we establish relationships for the constant term and the linear coefficient.

Analyze the factors of

-12

to find the integer values of

a

and

b

that maximize the linear term coefficient.

Choosing

b = 12

and

a = -1

gives

ab = -12

and results in

k = 2(12) + (-1) = 23

To maximize

2b + a

, we choose the largest possible positive factor of

-12

for the term multiplied by 2, which is 12, paired with the corresponding negative factor

-1

for

a

Anahtar Kavram

Factoring quadratic trinomials of the form

ax^2 + bx + c

with

a > 1

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