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Chapter 0: Problem 35

Multiply and simplify. $$ \left(4 u^{2}-5 v^{2}\right)\left(2 u^{2}+3 v^{2}\right) $$

Short Answer

Expert verified
The product and simplified form is \ \ \[8u^4 + 2u^2v^2 - 15v^4\]

Step by step solution

01

- Expand the Expression

Use the distributive property to expand \ \ \[ (4u^2 - 5v^2)(2u^2 + 3v^2) \ \] Multiply each term in the first parentheses by each term in the second parentheses: \ \ \[ (4u^2)(2u^2) + (4u^2)(3v^2) - (5v^2)(2u^2) - (5v^2)(3v^2) \]
02

- Perform Multiplication

Multiply the terms: \ \ \[ 4u^2 \times 2u^2 = 8u^4 \ \4u^2 \times 3v^2 = 12u^2v^2 \ \-5v^2 \times 2u^2 = -10u^2v^2 \ \-5v^2 \times 3v^2 = -15v^4 \]
03

- Combine Like Terms

Add the results from the multiplication: \ \ \[ 8u^4 + 12u^2v^2 - 10u^2v^2 - 15v^4 \ \] Combine like terms: \ \ \[ 8u^4 + 2u^2v^2 - 15v^4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
The distributive property is fundamental when multiplying polynomials. It states that you can distribute each term of one polynomial into every term of another polynomial. This is similar to multiplying numbers separately and then adding the products. For instance, in our original problem, we distribute each term of \((4u^{2}-5v^{2})\) to each term of \((2u^{2}+3v^{2})\). Here鈥檚 how it looks: \[ (4u^2)(2u^2) + (4u^2)(3v^2) - (5v^2)(2u^2) - (5v^2)(3v^2) \] Through this, we ensure every term in the first polynomial multiplies with every term in the second polynomial, giving us a new array of terms.
combining like terms
Once we distribute and multiply terms, the next step is combining like terms. Like terms are terms that have the same variable raised to the same power. For example, \(12u^2v^2\) and \(-10u^2v^2\) are like terms because their variables and exponents match. In our expanded polynomial: \[ 8u^4 + 12u^2v^2 - 10u^2v^2 - 15v^4 \] we combine \(12u^2v^2\) and \(-10u^2v^2\) to simplify: \[ 12u^2v^2 - 10u^2v^2 = 2u^2v^2 \] The reorganized and simplified polynomial now becomes: \[ 8u^4 + 2u^2v^2 - 15v^4\]
simplifying polynomials
Simplifying polynomials is the process of making the expression as concise as possible. This often involves distributing terms, combining like terms, and ensuring all coefficients and exponents are simplified. In our given problem, after correctly applying the distributive property and combining like terms, our polynomial becomes: \[ 8u^4 + 2u^2v^2 - 15v^4 \] Each term is distinct and there are no more like terms to combine. This is our simplest form of the polynomial. Simplification helps in solving and analyzing polynomial expressions efficiently.

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Most popular questions from this chapter

Perform the indicated operations and simplify. $$ (5 m-3)^{2} $$

Explain the similarity in simplifying the given expressions. a. \((3 x+2)(4 x-7)\) b. \((3 \sqrt{x}+\sqrt{2})(4 \sqrt{x}-\sqrt{7})\)

Use a calculator to approximate the expression to 2 decimal places. $$\frac{-3+5 \sqrt{2}}{7}$$

Simplify each expression. Assume that \(m\) and \(n\) are integers and that \(x\) and \(y\) are nonzero real numbers. \(\frac{x^{4 m-3} y^{5 n+7}}{x^{m-7} y^{3 n+2}}\)

A polynomial in the variable \(x\) is defined as an expression of the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\cdots+a_{1} x+a_{0} .\) Explain what this means.
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