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Chapter 5: Problem 20

Find each product. $$ -7 y\left(3+5 y^{2}-2 y^{3}\right) $$

Short Answer

Expert verified
\(-21 y - 35 y^3 + 14 y^4\).

Step by step solution

01

Distribute the Constant

Distribute \(-7 y\) to each term inside the parentheses\(-7 y \times 3 + -7 y \times 5 y^2 + -7 y \times -2 y^3\).
02

Calculate the Products

Perform the multiplication for each term: \(\-7 y \times 3 = -21 y\), \(-7 y \times 5 y^2 = -35 y^3\), \(-7 y \times -2 y^3 = 14 y^4\).
03

Combine the Results

Combine all the terms together: \(-21 y - 35 y^3 + 14 y^4\).

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Key Concepts

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

polynomial multiplication
Polynomial multiplication is the process of multiplying each term in one polynomial by every term in another polynomial. This is essential in algebra as it helps to simplify expressions and solve equations. In the given exercise, we multiply \(-7y\) by each term inside the parentheses \(3 + 5y^{2} - 2y^{3}\). Here's how it works: \(-7y \times 3, -7y \times 5y^{2}, \) and \(-7y \times -2y^{3}\).
  • First, perform each multiplication separately.
  • Then, combine all the resulting terms to get the final expression.

Each multiplication step is also managed by applying the distributive property which is fundamental to polynomial multiplication.
like terms
Like terms are terms in an algebraic expression that have the same variables raised to the same powers. Combining like terms simplifies our expression and makes it easier to work with. In our exercise, the terms \(-21y, -35y^3,\) and \(+14y^4\)
  • We need to ensure that each term is simplified correctly before adding.
  • Although they have different powers of \(y,\) so they can't be combined any further.

Remember, only like terms with the same variable and exponent can be combined.
exponents in algebra
Understanding exponents is crucial when performing algebraic operations. Exponents denote how many times a number (or variable) is multiplied by itself. For example, \(y^2\) means \(y \times y\). When multiplying variables with exponents, you add the exponents. In our solution, when multiplying \(-7y \times 5y^2\), we add the exponents of \(y\), which results in \(-35y^3\).
  • Always ensure to keep the base same and just add the exponents.
  • Negative exponents represent division, and zero exponents always equal one.

Keep these rules in mind while solving polynomial multiplications or any algebraic expressions involving exponents.

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Most popular questions from this chapter

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ 3 p^{3}\left(2 p^{2}+5 p\right)\left(p^{3}+2 p+1\right) $$

List all positive integer factors of each number. $$ 48 $$

Perform each division using the "long division" process. $$ \frac{6 p^{4}-16 p^{3}+15 p^{2}-5 p+10}{3 p+1} $$

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 20 \frac{1}{2} \times 19 \frac{1}{2} $$

Multiply. $$ 5(x+4) $$
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