Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.
Table of contents
- 0. Review of Algebra4h 18m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations1h 43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 5m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 22m
- 10. Combinatorics & Probability1h 45m
0. Review of Algebra
Multiplying Polynomials
Problem 8
Textbook Question
Perform the indicated operations. -2x3(x4-8)
Verified step by step guidance1
Identify the expression to simplify: \(-2x^3(x^4 - 8)\).
Apply the distributive property, which means multiplying \(-2x^3\) by each term inside the parentheses separately.
Multiply \(-2x^3\) by \(x^4\): use the rule of exponents \(x^a \cdot x^b = x^{a+b}\) to combine the powers of \(x\).
Multiply \(-2x^3\) by \(-8\): multiply the coefficients and keep the variable part as is.
Write the simplified expression by combining the results from the two multiplications.
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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 raised to whole-number exponents and coefficients. Understanding how to identify terms, coefficients, and exponents is essential for manipulating and simplifying polynomials.
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Distributive Property
The distributive property states that multiplying a single term by a sum or difference inside parentheses involves multiplying the term by each addend separately. This property is key to expanding expressions like -2x^3(x^4 - 8).
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Laws of Exponents
Laws of exponents govern how to multiply powers with the same base by adding their exponents. For example, when multiplying x^3 by x^4, the result is x^(3+4) = x^7. This rule is crucial for simplifying the product in the given expression.
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