/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Find each product. $$ \left(... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 39

Find each product. $$ \left(6 x^{4}-4 x^{2}+8 x\right)\left(\frac{1}{2} x+3\right) $$

Short Answer

Expert verified
The product is \(3x^{5} + 18x^{4} - 2x^{3} - 8x^{2} + 24x\).

Step by step solution

01

- Distribute Each Term

To find the product \( \big(6 x^{4}-4 x^{2}+8 x\big)\big(\frac{1}{2} x+3\big) \) use the distributive property. Distribute each term in the first polynomial to each term in the second polynomial.
02

- Multiply Each Pair of Terms

Multiply each pair of terms: \(6x^{4} \times \frac{1}{2}x\), \(6x^{4} \times 3\), \(-4x^{2} \times \frac{1}{2}x\), \(-4x^{2} \times 3\), \(8x \times \frac{1}{2}x\), \(8x \times 3\): \(6x^{4} \times \frac{1}{2}x = 3x^{5}\) \(6x^{4} \times 3 = 18x^{4}\) \(-4x^{2} \times \frac{1}{2}x = -2x^{3}\) \(-4x^{2} \times 3 = -12x^{2}\) \(8x \times \frac{1}{2}x = 4x^{2}\) \(8x \times 3 = 24x\)
03

- Combine Like Terms

Combine the like terms from the result of the multiplication: \(3x^{5}\), \(18x^{4}\), \(-2x^{3}\), \(-12x^{2} + 4x^{2}\), \(24x\): \(3x^{5} + 18x^{4} - 2x^{3} - 8x^{2} + 24x\)

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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 essential in polynomial multiplication. It allows us to multiply each term in one polynomial by every term in the other polynomial. Think of it as distributing or spreading out each term. For example, in our exercise \((6x^4 - 4x^2 + 8x)(\frac{1}{2}x + 3)\), we need to distribute every term in \(6x^4 - 4x^2 + 8x\) with both terms in \(\frac{1}{2}x + 3\). This involves multiple smaller multiplications which can be managed step by step.

combining like terms
After using the distributive property, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power. This process helps in simplifying the polynomial to its simplest form.

For instance, in the exercise, after distributing we get terms like \(3x^5\), \(-2x^3\), \(-12x^2\), and \(+4x^2\). To simplify, we combine the \(-12x^2\) and \(+4x^2\) to get \(-8x^2\). This results in the final expression \(3x^5 +18x^4 - 2x^3 - 8x^2 + 24x\).

algebraic expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. Polynomial expressions are specific types of algebraic expressions that involve terms with variables raised to whole-number exponents.

In our example, \(6x^4 - 4x^2 + 8x\) and \(\frac{1}{2}x + 3\) are polynomial expressions. Understanding each component of these expressions is crucial for performing operations like multiplication. Always make sure you can identify the coefficients (numeric factors) and the variables in each term.

exponents
Exponents are used in algebra to denote repeated multiplication of a number or variable by itself. They are written as a superscript number next to the base number or variable. For example, in \(x^4\), 4 is the exponent and it indicates \(x\) is multiplied by itself 4 times.

When multiplying terms with exponents, you add the exponents if the bases are the same. For instance, \(x^a \times x^b = x^{a+b}\). In our example, \(6x^4 \times \frac{1}{2}x = 3x^5\) because the exponents 4 and 1 (from \(x^1\)) are added to get 5.

Understanding this rule of exponents is pivotal for accurately performing polynomial multiplication and simplifying the resulting expressions.

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