Examkin

Soru

Zorluk: Çok zorFactoring Polynomials

An algebraic expression of the form x4−10x2y2+9y4−x2+9y2x^4 - 10x^2y^2 + 9y^4 - x^2 + 9y^2 is defined for all real numbers xx and yy. When this expression is factored completely over the integers, which of the following is one of its factors?

  1. A
    x−9y2x - 9y^2
  2. B
    x−y−1x - y - 1
  3. C
    x−yx - y
  4. x+3yx + 3yCevap
  5. E
    x2−y2+1x^2 - y^2 + 1

Cevap

The binomial x+3yx + 3y is a factor of the expression.
To factor the polynomial completely, we group the terms as (x4−10x2y2+9y4)−(x2−9y2)(x^4 - 10x^2y^2 + 9y^4) - (x^2 - 9y^2). The first trinomial is in quadratic form and factors to (x2−9y2)(x2−y2)(x^2 - 9y^2)(x^2 - y^2). The expression is then rewritten as (x2−9y2)(x2−y2)−(x2−9y2)(x^2 - 9y^2)(x^2 - y^2) - (x^2 - 9y^2), which has a GCF of (x2−9y2)(x^2 - 9y^2). Factoring out the GCF yields (x2−9y2)(x2−y2−1)(x^2 - 9y^2)(x^2 - y^2 - 1). Finally, the difference of squares (x2−9y2)(x^2 - 9y^2) factors into (x−3y)(x+3y)(x - 3y)(x + 3y). The fully factored expression is (x−3y)(x+3y)(x2−y2−1)(x - 3y)(x + 3y)(x^2 - y^2 - 1), which contains the factor x+3yx + 3y.

Adım Adım Çözüm

1
Group the terms of the polynomial into two parts.
(x4−10x2y2+9y4)−(x2−9y2)(x^4 - 10x^2y^2 + 9y^4) - (x^2 - 9y^2)
Grouping terms allows us to find and extract common algebraic factors from distinct parts of the polynomial.
2
Factor the trinomial from the first group as a quadratic form in terms of x2x^2 and y2y^2.
(x2−9y2)(x2−y2)(x^2 - 9y^2)(x^2 - y^2)
The trinomial x4−10x2y2+9y4x^4 - 10x^2y^2 + 9y^4 can be written as (x2)2−10(x2)(y2)+9(y2)2(x^2)^2 - 10(x^2)(y^2) + 9(y^2)^2, which factors as (x2−9y2)(x2−y2)(x^2 - 9y^2)(x^2 - y^2).
3
Substitute this factorization back into the grouped expression and factor out the greatest common factor.
(x2−9y2)(x2−y2−1)(x^2 - 9y^2)(x^2 - y^2 - 1)
Both parts of the grouped expression share (x2−9y2)(x^2 - 9y^2) as a GCF, leaving (x2−y2−1)(x^2 - y^2 - 1) when factored out.
4
Factor the difference of squares term completely over the integers.
(x−3y)(x+3y)(x2−y2−1)(x - 3y)(x + 3y)(x^2 - y^2 - 1)
The term (x2−9y2)(x^2 - 9y^2) is a difference of squares of the form a2−b2a^2 - b^2, which factors into (a−b)(a+b)(a - b)(a + b).

Anahtar Kavram

Factoring high-degree polynomials using quadratic form substitution, grouping, and difference of squares.
Bu soruyu puanla

Konular

Factoring Polynomials

Bu konudan daha fazla soru