Find each product. (2z4-3y)2

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 50

Chapter 1, Problem 50

Find each product. (2z⁴-3y)²

Verified step by step guidance

Recognize that the expression is a binomial squared: $(2z^4 - 3y)^2$. This means you will use the formula for the square of a binomial: $(a - b)^2 = a^2 - 2ab + b^2$.

Identify the terms: let $a = 2z^4$ and $b = 3y$.

Calculate the square of the first term: $a^2 = (2z^4)^2 = 2^2 \cdot (z^4)^2 = 4z^{8}$.

Calculate twice the product of the two terms with a negative sign: $-2ab = -2 \cdot (2z^4) \cdot (3y) = -12z^4 y$.

Calculate the square of the second term: $b^2 = (3y)^2 = 9y^2$. Then, combine all parts to write the expanded expression: $4z^{8} - 12z^4 y + 9y^2$.

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Key Concepts

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

Polynomial Expressions

Polynomial expressions are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. Understanding the structure of polynomials helps in identifying terms and applying operations like expansion or factoring.

Binomial Squaring Formula

The binomial squaring formula states that (a - b)^2 = a^2 - 2ab + b^2. This formula is essential for expanding the square of a binomial expression, allowing you to rewrite it as a trinomial by squaring each term and doubling the product of the two terms.

Exponent Rules

Exponent rules govern how to handle powers of variables, such as multiplying powers by adding exponents and raising a power to another power by multiplying exponents. These rules are crucial when expanding expressions like (2z^4 - 3y)^2 to correctly simplify terms.

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Textbook Question

Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. A ⊆ U

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Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√x² • ⁵√x³

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Multiply or divide as indicated. Write answers in lowest terms as needed. $2 and one half divided by 1 and five sevenths$

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Factor each trinomial, if possible. See Examples 3 and 4. $24 a^{4} + 10 a^{3} b - 4 a^{2} b^{2}$

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