/*! 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 59 Multiply and check. $$ \left... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 59

Multiply and check. $$ \left(y^{2}-7\right)\left(2 y^{3}+y+1\right) $$

Short Answer

Expert verified
The result is \(2y^{5} - 13y^{3} + y^{2} - 7y - 7\).

Step by step solution

01

Distribute each term

Distribute each term in \(\right(y^{2}-7\right)\) to each term in \(2y^{3}+y+1\). This means you'll multiply \(y^{2}\) by \(2y^{3}\), \(y^{2}\) by \(y\), \(y^{2}\) by \(1\), and do the same for \(-7\).
02

Multiply the terms

Compute the products from Step 1:\( y^{2} \times 2y^{3} = 2y^{5} \, \ y^{2} \times y = y^{3} \, \ y^{2} \times 1 = y^{2} \, \ -7 \times 2y^{3} = -14y^{3} \, \ -7 \times y = -7y \, \ -7 \times 1 = -7\).
03

Combine like terms

Combine the like terms from the products:\(2y^{5} \, \ y^{3} -14y^{3} = -13y^{3} \, \ y^{2} \, \ -7y \, \ -7\). The final polynomial is \(2y^{5} - 13y^{3} + y^{2} - 7y - 7\).
04

Check the result

Verify each distribution and combination by checking each term carefully. Ensure no terms are missed or incorrectly combined.

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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 a fundamental concept in algebra. It states that for any numbers or expressions, distributing a factor across terms inside parentheses is always valid. In mathematical form, this is shown as: $$a(b+c) = ab + ac$$
In this problem, we use the distributive property to multiply each term inside the first polynomial \((y^2 - 7)\) by each term inside the second polynomial \((2y^3 + y + 1)\). This will give us individual products that we can work with.
Combining Like Terms
Combining like terms is a crucial step in simplifying polynomial expressions. Like terms are terms that have the same variables raised to the same powers. For example, in the expression \(2y^5 + y^3 -14y^3 + y^2 - 7y - 7\), the terms \y^3\ and \-14y^3\ are like terms because they both contain \y^3\.
To combine like terms, simply add or subtract their coefficients. For example: \ y^3 - 14y^3 = -13y^3 \. Always ensure no terms are missed or incorrectly combined to keep the polynomial accurate.
Monomials
A monomial is a single term algebraic expression consisting of one variable with a coefficient. In \(2y^5 - 13y^3 + y^2 - 7y - 7\), each individual term is a monomial. The term \2y^5\ is considered a monomial because it is a single variable term with a numerical coefficient. Each term in the polynomial can be a monomial before combining like terms.
Understanding monomials helps in breaking down complex polynomials into manageable parts, making it easier to apply other algebraic operations.
Exponents
Exponents are mathematical notations indicating the number of times a number (the base) is multiplied by itself. In the polynomial expression \(2y^5 - 13y^3 + y^2 - 7y - 7\), exponents like 5 in \2y^5\ show that \y\ is multiplied by itself five times.
When multiplying expressions with the same base, add their exponents. For example, \(y^2 \times 2y^3 = 2y^{2+3} = 2y^5\). Mastering exponents helps in simplifying and manipulating polynomial terms effectively.

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Most popular questions from this chapter

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{a^{3}}{-2 b^{5}}\right)^{4} $$

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Find a value of the variable that shows that the two expressions are not equivalent. Answers may vary. $$ \frac{t^{6}}{t^{2}} ; t^{3} $$
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