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Chapter 5: Problem 41

Multiply. See Example 7. $$ \left(x^{2}+y\right)\left(4 x-y^{4}\right) $$

Short Answer

Expert verified
The result is \(4x^3 - x^2y^4 + 4xy - y^5\).

Step by step solution

01

Apply Distributive Property

To multiply the binomials, we'll use the distributive property (also known as the FOIL method for binomials). This process involves multiplying each term in the first binomial by each term in the second binomial.
02

Expand the Binomials

First, multiply the first term of the first binomial by both terms of the second binomial:\[(x^2 + y)(4x - y^4) = x^2(4x - y^4) + y(4x - y^4)\]
03

Multiply Each Term

Perform the multiplications separately:- \(x^2 \cdot 4x = 4x^3\)- \(x^2 \cdot (-y^4) = -x^2 y^4\)- \(y \cdot 4x = 4xy\)- \(y \cdot (-y^4) = -y^5\)
04

Combine All Terms

Now, combine the results of the multiplications:\[4x^3 - x^2y^4 + 4xy - y^5\]

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Key Concepts

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

Understanding Binomials
A binomial is a type of polynomial with exactly two terms. The typical structure of a binomial is simple: it's an algebraic expression that includes variables, numbers, or both, separated by either a plus or a minus sign.
Here's how to identify key parts of a binomial:
  • Each term can consist of a constant (a number), a variable (like \(x\) or \(y\)), or a combination (like \(3x^2\)).
  • Terms are separated by addition or subtraction, forming the structure such as \(a + b\).
In the given exercise, \(x^2 + y\) and \(4x - y^4\) are both examples of binomials, characterized by having two distinct terms each.
Understanding binomials is essential to polynomial multiplication, leading us to our next topic.
Polynomial Multiplication Techniques
Polynomial multiplication involves multiplying two or more polynomials together. This process leverages the distributive property to simplify the operation. When multiplying binomials, each term of one polynomial is multiplied by each term of the other. This is similar to the step-by-step distributing each term separately:
  • First, take each term from the first polynomial.
  • Multiply it by each term in the second polynomial.
  • The result is a new polynomial, typically with more terms.
For our exercise, multiplying the binomials \((x^2 + y)(4x - y^4)\) leads to the expression \(x^2(4x - y^4) + y(4x - y^4)\).
Each item is calculated individually, and this method ensures that all combinations are accounted for in the result.
Using the FOIL Method for Binomials
The FOIL method is a quick way to multiply two binomials. It's an acronym that stands for _First, Outer, Inner, Last_. Each refers to the pair of terms you multiply together in each step. Here's a breakdown of how it works:
  • **First:** Multiply the first terms in each binomial.
  • **Outer:** Multiply the outermost terms in the product.
  • **Inner:** Multiply the innermost terms between the two binomials.
  • **Last:** Multiply the last terms in each binomial.
Applying FOIL to our exercise won't exactly match the acronym due to the nature of the polynomial terms, but the idea remains to systematically multiply each term
In our case, combining all steps leads to the polynomial expressions: \(4x^3 - x^2y^4 + 4xy - y^5\). This neat trick ensures none of the multiplicative combinations are missed.

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Most popular questions from this chapter

Divide. $$ \frac{x^{5}+x^{2}}{x^{2}+x} $$

The volume of the swimming pool shown is \(\left(36 x^{5}-12 x^{3}\right.\) \(\left.+6 x^{2}\right)\) cubic feet. If its height is \(2 x\) feet and its width is \(3 x\) feet, find its length.

Multiply. See Section 5.1 $$ \left(12 x^{3}\right)\left(-x^{5}\right) $$

Simplify each expression. See Section \(5.1 .\) $$ \frac{-48 a b^{6}}{32 a b^{3}} $$

Multiply. $$(a+4)\left(a^{2}-6 a+6\right)$$
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