Factor each trinomial, if possible. See Examples 3 and 4. 4x2y2+28xy+49

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 55

1장, 문제 55

Factor each trinomial, if possible. See Examples 3 and 4. 4x²y²+28xy+49

검증된 단계별 안내

Identify the trinomial to factor: $4x^{2}y^{2} + 28xy + 49$.

Recognize that this trinomial resembles a perfect square trinomial of the form $a^{2} + 2ab + b^{2}$, where $a$ and $b$ are expressions.

Find $a$ by taking the square root of the first term: $\sqrt{4x^{2}y^{2}} = 2xy$.

Find $b$ by taking the square root of the last term: $\sqrt{49} = 7$.

Check if the middle term matches \$2ab$: calculate \(2 \times 2xy \times 7 = 28xy$, which matches the middle term, confirming the trinomial factors as \)(2xy + 7)^{2}$.

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Factoring Trinomials

Factoring trinomials involves rewriting a quadratic expression as a product of two binomials. This process often requires identifying two numbers that multiply to the constant term and add to the coefficient of the middle term. Recognizing patterns and applying factoring techniques simplifies expressions and solves equations.

Greatest Common Factor (GCF)

The Greatest Common Factor is the largest factor shared by all terms in an expression. Factoring out the GCF first simplifies the trinomial, making it easier to factor further. In expressions with variables, the GCF includes the lowest powers of common variables.

Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial, typically in the form a² + 2ab + b². Recognizing this pattern allows quick factoring into (a + b)² or (a - b)². This is useful when the first and last terms are perfect squares and the middle term is twice the product of their roots.