/*! 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 36 Find each product. $$ (2 a+3... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 36

Find each product. $$ (2 a+3)\left(a^{4}-a^{3}+a^{2}-a+1\right) $$

Short Answer

Expert verified
2a^{5} + a^{4} - a^{3} + a^{2} - a + 3

Step by step solution

01

Distribute the first term

Multiply each term inside the parentheses \(a^{4}-a^{3}+a^{2}-a+1\) by the first term \(2a\). Use the distributive property to expand: \(2a(a^{4}-a^{3}+a^{2}-a+1) = 2a \times a^{4} - 2a \times a^{3} + 2a \times a^{2} - 2a \times a + 2a \times 1\)This simplifies to \(2a^{5} - 2a^{4} + 2a^{3} - 2a^{2} + 2a\).
02

Distribute the second term

Multiply each term inside the parentheses \(a^{4}-a^{3}+a^{2}-a+1\) by the second term \(3\). Use the distributive property to expand: \(3(a^{4}-a^{3}+a^{2}-a+1) = 3 \times a^{4} - 3 \times a^{3} + 3 \times a^{2} - 3 \times a + 3 \times 1\)This simplifies to \(3a^{4} - 3a^{3} + 3a^{2} - 3a + 3\).
03

Combine like terms

Add up all the terms from Step 1 and Step 2: \(2a^{5} - 2a^{4} + 2a^{3} - 2a^{2} + 2a + 3a^{4} - 3a^{3} + 3a^{2} - 3a + 3\).Now, combine like terms to simplify the expression: \2a^{5} + ( - 2a^{4} + 3a^{4}) + (2a^{3} - 3a^{3}) + (-2a^{2} + 3a^{2}) + (2a - 3a) + 3\This results in \2a^{5} + a^{4} - a^{3} + a^{2} - a + 3\.

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Key Concepts

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

polynomial multiplication
Polynomial multiplication involves multiplying each term in one polynomial by every term in the other polynomial. This process ensures that no terms are omitted. In the exercise provided, we multiplied the binomial (2a + 3) with a polynomial of five terms (a^4 - a^3 + a^2 - a + 1).
This means we first distributed the term 2a to every term inside the parentheses, which required using the distributive property: 2a(a^4) - 2a(a^3) + 2a(a^2) - 2a(a) + 2a(1).
Then, we repeated this process for the term 3 independently: 3(a^4) - 3(a^3) + 3(a^2) - 3(a) + 3(1).
like terms
Like terms are terms in an algebraic expression that have identical variable parts, regardless of the coefficients they carry. For example, in the multiplication step, terms like a^4d and a^4d are like terms.
Variable parts must match for terms to be considered 'like'. This is why, after distributing, we combine like terms in the expression: 2a^5d and 3a^4.
Precisely, the terms -2a^4 and 3a^4 are combined to make a^4. Similarly, 2a^3 and -3a^3 combine to -a^3.
simplifying expressions
Simplifying expressions is the process of combining like terms into a single term to make the expression concise and easier to manage. In the final step of solving the exercise, we simplified the distributed terms by combining all like terms: 2a^5 - 2a^4 + 2a^3 - 2a^2 + 2a and 3a^4 - 3a^3 + 3a^2 - 3a + 3.
We carefully lined up terms with the same powers and combined them: 2a^5d stays as is, because there is no other term with a^5. However, -2a^4 combined with 3a^4to become a^4.
This method of combining ensures that our final expression is as simplified and as clear as possible: 2a^5 + a^4 - a^3 + a^2 - a + 3.

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