Examkin

Soru

Zorluk: ZorFactoring Polynomials

For all real values of xx and yy, the expression 4x2−9y2−12y−44x^2 - 9y^2 - 12y - 4 can be factored into the form (2x+ay+b)(2x−cy−d)(2x + ay + b)(2x - cy - d), where a,b,ca, b, c, and dd are positive integers. What is the value of a+b+c+da + b + c + d?

Cevap: 10

Cevap

The value of a+b+c+da + b + c + d is 10.
Grouping the yy terms gives 4x2−(9y2+12y+4)4x^2 - (9y^2 + 12y + 4). Factoring the quadratic within the parentheses gives 4x2−(3y+2)24x^2 - (3y + 2)^2. Applying the difference of squares identity (u2−v2)=(u+v)(u−v)(u^2 - v^2) = (u + v)(u - v) leads to (2x+(3y+2))(2x−(3y+2))=(2x+3y+2)(2x−3y−2)(2x + (3y + 2))(2x - (3y + 2)) = (2x + 3y + 2)(2x - 3y - 2). Matching this expression to the template (2x+ay+b)(2x−cy−d)(2x + ay + b)(2x - cy - d) reveals that a=3a = 3, b=2b = 2, c=3c = 3, and d=2d = 2, all of which are positive integers. The sum of these values is 3+2+3+2=103 + 2 + 3 + 2 = 10.

Adım Adım Çözüm

1
Group the terms containing yy
4x2−(9y2+12y+4)4x^2 - (9y^2 + 12y + 4)
Grouping the terms allows us to identify a perfect square trinomial pattern.
2
Factor the trinomial inside the parentheses
4x2−(3y+2)24x^2 - (3y + 2)^2
The expression 9y2+12y+49y^2 + 12y + 4 is a perfect square trinomial of the form (3y)2+2(3y)(2)+22(3y)^2 + 2(3y)(2) + 2^2.
3
Apply the difference of squares identity
(2x+(3y+2))(2x−(3y+2))(2x + (3y + 2))(2x - (3y + 2))
The expression is in the form u2−v2u^2 - v^2, where u=2xu = 2x and v=3y+2v = 3y + 2. Using u2−v2=(u+v)(u−v)u^2 - v^2 = (u + v)(u - v) factors the expression.
4
Simplify the factored binomials by distributing signs
(2x+3y+2)(2x−3y−2)(2x + 3y + 2)(2x - 3y - 2)
Simplifying the expressions removes inner parentheses, allowing comparison with the target template.
5
Compare with the template (2x+ay+b)(2x−cy−d)(2x + ay + b)(2x - cy - d) to find the constants
a=3a = 3, b=2b = 2, c=3c = 3, d=2d = 2
Matching the terms directly gives ay=3y⇒a=3ay = 3y \Rightarrow a = 3, b=2b = 2, −cy=−3y⇒c=3-cy = -3y \Rightarrow c = 3, and −d=−2⇒d=2-d = -2 \Rightarrow d = 2.
6
Calculate the sum of the positive integers
10
The question asks for the value of a+b+c+da + b + c + d, which is 3+2+3+2=103 + 2 + 3 + 2 = 10.

Anahtar Kavram

Factoring polynomials using grouping, perfect square trinomials, and the difference of squares identity
Tahmini Süre:2m 0s
Bu soruyu puanla

Konular

Factoring Polynomials

Bu konudan daha fazla soru