In Exercises 103–114, factor completely. x4−5x2y2+4y4

Ch. P - Fundamental Concepts of Algebra

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

All textbooks

Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 109

Chapter 1, Problem 109

In Exercises 103–114, factor completely. x⁴−5x²y²+4y⁴

Verified step by step guidance

Recognize that the given expression $ x^4 - 5x^2y^2 + 4y^4 $ is a quadratic form in terms of $ x^2 $ and $ y^2 $. Rewrite it as $ (x^2)^2 - 5(x^2)(y^2) + 4(y^2)^2 $.

Observe that this is a trinomial in the form $ a^2 - 2ab + b^2 $, which can potentially be factored as a product of two binomials. Let $ u = x^2 $ and $ v = y^2 $, so the expression becomes $ u^2 - 5uv + 4v^2 $.

Factor the trinomial $ u^2 - 5uv + 4v^2 $ by finding two numbers that multiply to $ 4 $ (the constant term) and add to $ -5 $ (the coefficient of $ uv $). These numbers are $ -4 $ and $ -1 $.

Rewrite the trinomial as $ (u - 4v)(u - v) $, substituting back $ u = x^2 $ and $ v = y^2 $. This gives $ (x^2 - 4y^2)(x^2 - y^2) $.

Notice that both $ x^2 - 4y^2 $ and $ x^2 - y^2 $ are differences of squares. Factor them further as $ (x - 2y)(x + 2y) $ and $ (x - y)(x + y) $, respectively. The fully factored form is $ (x - 2y)(x + 2y)(x - y)(x + y) $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process is essential for simplifying expressions and solving equations. In the case of the given polynomial, recognizing patterns such as the difference of squares or perfect square trinomials can aid in the factoring process.

Quadratic Form

The expression x^4−5x^2y^2+4y^4 can be viewed as a quadratic in terms of x^2. By substituting u = x^2, the polynomial transforms into a standard quadratic form, making it easier to apply factoring techniques. This approach allows for the identification of roots and factors more straightforwardly.

Difference of Squares

The difference of squares is a factoring technique used when an expression can be written in the form a^2 - b^2, which factors into (a - b)(a + b). In the context of the given polynomial, recognizing components that fit this pattern can simplify the factoring process and lead to a complete factorization of the expression.

Recommended video:

06:24

Solving Quadratic Equations by Completing the Square

Related Practice

Textbook Question

Simplify by reducing the index of the radical. $\sqrt[12]{x^{4} y^{8}}$

Textbook Question

Simplify each rational expression. Also, list all numbers that must be excluded from the domain. [x^3+2x^2]/[x+2]

Textbook Question

Insert either <, >, or = in the shaded area to make a true statement. $\frac{30}{40}\) - $\frac{3}{4}$ \ $\square$ \ $\frac{14}{15}$ $\cdot$ \(\frac{15}{14}$

Textbook Question

In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. 8/13÷8/13 □ |−1|

Textbook Question

Factor and simplify each algebraic expression. $[12 x^{- \frac{1}{2}} + 6 x^{- \frac{3}{2}}]$

views

Textbook Question

In Exercises 111–114, simplify each expression. Assume that all variables represent positive numbers. (49x−2y⁴)^−1/2(xy^1/2)

views