Decompose the quadratic trinomial 6x^2 - 11x - 10 into a product of two binomial factors s… — Armed Services Vocational Aptitude Battery (ASVAB)

Decompose the quadratic trinomial $6x^2 - 11x - 10$ into a product of two binomial factors structured as $(ax - b)(cx + d)$ . Assume all parameters ( $a$ , $b$ , $c$ , and $d$ ) are strictly positive integers. To ensure a unique arrangement, the leading coefficient $a$ must be strictly less than $c$ . Provide the values of the four parameters.

Cevap:

a =

【2】,

b =

【5】,

c =

【3】, and

d =

【2】.

Cevap

The parameters are

a = 2

b = 5

c = 3

, and

d = 2

Factoring the polynomial

6x^2 - 11x - 10

using the AC method produces

(2x - 5)(3x + 2)

. Matching these two factors against the template

(ax - b)(cx + d)

under the rule that

a < c

(since

2 < 3

) perfectly determines the positive integers as

a = 2

b = 5

c = 3

, and

d = 2

Adım Adım Çözüm

Identify the factoring technique for a non-monic quadratic trinomial in the form

Ax^2 + Bx + C

The AC method (factoring by grouping) is required because the leading coefficient

A = 6

is greater than 1.

Directly finding factors of the constant

C

that sum to

B

only works when

A = 1

Multiply the leading coefficient

A

by the constant term

C

6 \times (-10) = -60

This establishes the target product needed to properly split the middle term for grouping.

Find two integers that multiply to

-60

and add to the middle coefficient

B = -11

The correct pair of integers is

-15

and

4

(-15) \times 4 = -60

and

-15 + 4 = -11

Rewrite the middle term using these two integers and factor the resulting four-term expression by grouping.

6x^2 - 15x + 4x - 10 = 3x(2x - 5) + 2(2x - 5) = (2x - 5)(3x + 2)

Grouping the first two terms and the last two terms reveals a common binomial factor of

(2x - 5)

Match the factored expression to the required template

(ax - b)(cx + d)

while adhering to the condition

a < c

The factors are

(2x - 5)

and

(3x + 2)

. Since

2 < 3

(ax - b)

corresponds exactly to

(2x - 5)

, yielding

a=2

and

b=5

. Consequently,

(cx + d)

corresponds to

(3x + 2)

, yielding

c=3

and

d=2

This guarantees a unique, correctly-ordered set of strictly positive parameters.

Anahtar Kavram

Factoring non-monic quadratic trinomials via grouping

Soru

Cevap

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