If the expression \frac{6x^3 + 7x^2 - 14x + 19}{2x + 5} is equivalent to ax^2 + bx + c + \… — SAT

If the expression $\frac{6x^3 + 7x^2 - 14x + 19}{2x + 5}$ is equivalent to $ax^2 + bx + c + \frac{k}{2x + 5}$ for all $x \neq -2.5$ , where $a$ , $b$ , $c$ , and $k$ are constants, what is the value of $a - b + c + k$ ?

Cevap: 14

Cevap

Dividing the numerator

6x^3 + 7x^2 - 14x + 19

by the denominator

2x + 5

using polynomial division yields a quotient of

3x^2 - 4x + 3

and a remainder of

4

. The equivalent expression is

3x^2 - 4x + 3 + \frac{4}{2x + 5}

. Comparing this with the form

ax^2 + bx + c + \frac{k}{2x + 5}

gives the values

a = 3

b = -4

c = 3

, and

k = 4

. Substituting these values into

a - b + c + k

gives

3 - (-4) + 3 + 4 = 14

Adım Adım Çözüm

Perform the first step of polynomial long division by dividing

6x^3

2x

Quotient term:

3x^2

; Remainder:

-8x^2 - 14x + 19

To eliminate the highest-degree term of the numerator.

Perform the second step of division by dividing

-8x^2

2x

Quotient term:

-4x

; Remainder:

6x + 19

To find the next term of the quotient.

Perform the third step of division by dividing

6x

2x

Quotient term:

3

; Remainder:

4

To find the constant term of the quotient and the final remainder.

Compare the resulting expression

3x^2 - 4x + 3 + \frac{4}{2x + 5}

with the given form to identify the constants

a

b

c

, and

k

a = 3

b = -4

c = 3

k = 4

To map the coefficients of equivalent algebraic expressions.

Evaluate the expression

a - b + c + k

using the identified values.

3 - (-4) + 3 + 4 = 14

To obtain the final numeric value requested.

Anahtar Kavram

Equivalent Algebraic Expressions via Polynomial Long Division

Soru

Cevap

Adım Adım Çözüm

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