/*! 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 48 Find each product of the monomia... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 48

Find each product of the monomial and the polynomial. $$4 y^{2}\left(5 y^{2}-6 y+3\right)$$

Short Answer

Expert verified
The product of the given monomial and polynomial is \(20 y^{4}-24 y^{3}+12 y^{2}\).

Step by step solution

01

Identify the monomial and the polynomial

In the given expression \(4 y^{2}(5 y^{2}-6 y+3)\), the monomial is \(4 y^{2}\) and the polynomial is \((5 y^{2}-6 y+3)\).
02

Apply the distributive property

Use the distributive property of multiplication over addition to multiply the monomial with each term of the polynomial. This means, you must multiply \(4 y^{2}\) by each of \(5 y^{2}\), \(-6 y\) and \(3\).
03

Carry out the multiplications

Doing the multiplications, you get \(4 y^{2} * 5 y^{2} = 20 y^{4}\), \(4 y^{2} * -6 y = -24 y^{3}\) and \(4 y^{2} * 3 = 12 y^{2}\).
04

Write the final result

So substituting these values back into the original equation gives us the final result as \(20 y^{4}-24 y^{3}+12 y^{2}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Monomials
In algebra, a monomial is an expression that consists of only one term. This term includes a constant, coefficients, and variables, which can be raised to a power. Monomials are among the simplest forms in the family of algebraic expressions. The given expression, \(4y^2(5y^2 - 6y + 3)\), features the monomial \(4y^2\). Here鈥檚 why:
  • The number \(4\) serves as the coefficient, which multiplies the variable part.
  • \(y^2\) represents the variable raised to a power, indicating that the variable is squared.
Recognizing the structure of a monomial helps with overall algebra manipulations. It is also the basis for employing other operations like distribution.Understanding these elements is crucial for more advanced math, where you will encounter polynomials or more complex algebraic expressions.
Exploring the Distributive Property
The distributive property is a vital principle in algebra. It allows you to simplify expressions by 'distributing' a multiplier over addition or subtraction inside parentheses. In our exercise, this property is used when multiplying a monomial by a polynomial.

Let's break down the process:
  • You multiply the monomial by each term inside the polynomial separately.
  • In our example, \(4y^2\) is multiplied by each term of the polynomial \((5y^2 - 6y + 3)\).
  • This results in three individual products: \(4y^2 \times 5y^2 = 20y^4\), \(4y^2 \times -6y = -24y^3\), and \(4y^2 \times 3 = 12y^2\).
The distributive property is powerful because it simplifies complex problems step by step. It breaks them into manageable parts, making them easier to solve and understand.Once you master this technique, handling more intricate algebraic expressions becomes more straightforward.
Working with Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They form the language of algebra. In the exercise, you start with a complex expression: \(4y^2(5y^2 - 6y + 3)\). To simplify this expression:
  • Understand that it includes both a monomial and a polynomial.
  • Apply the distributive property to multiply the monomial with the polynomial's terms.
  • This will yield the new expression \(20y^4 - 24y^3 + 12y^2\).
Each resultant term keeps its own power of \(y\) as well as its respective coefficients, which are calculated through multiplication.

Algebraic expressions often appear in equations, so becoming adept with them can pave the way to solving equations and making critical math connections. By recognizing their components, you'll strengthen your mathematics foundation.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve the system: $$\left\\{\begin{array}{l}7 x-6 y=17 \\\3 x+y=18\end{array}\right.$$ (Section 4.3, Example 2)

Solve each system by the method of your choice. $$\left\\{\begin{array}{l}3 x+4 y=7 \\\2 x+7 y=9\end{array}\right.$$

When dividing a polynomial by a binomial, explain when to stop dividing.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I divide monomials by dividing coefficients and subtracting exponents.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}+x}{x}=x$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.