/*! 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 60 Multiply the binomials. $$(z-2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 60

Multiply the binomials. $$(z-2)(4 z+3)$$

Short Answer

Expert verified
The product is \(4z^2 - 5z - 6\).

Step by step solution

01

Apply the Distributive Property

To multiply the binomials \((z-2)(4z+3)\), apply the distributive property by multiplying each term in the first binomial by each term in the second. Start with: \[ z imes (4z+3) \] and \[ -2 imes (4z+3) \]
02

Expand the Binomials

Calculate the individual products for each term:- \(z \times 4z = 4z^2\)- \(z \times 3 = 3z\)- \(-2 \times 4z = -8z\)- \(-2 \times 3 = -6\)
03

Combine Like Terms

Combine the like terms from the expanded expression:- The expression is \(4z^2 + 3z - 8z - 6\)- Combine the \(z\) terms: \(3z - 8z = -5z\).- Thus, the simplified result is: \(4z^2 - 5z - 6\).

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.

Distributive Property
The Distributive Property is a key mathematical principle that simplifies multiplication over addition or subtraction. When multiplying two binomials, this property comes to the rescue. In our example, \((z-2)(4z+3)\), we distribute each term in the first binomial to each term in the second binomial. First, distribute \(z\) over the second binomial:
  • \(z \times 4z\)
  • \(z \times 3\)
Next, do the same for \(-2\):
  • \(-2 \times 4z\)
  • \(-2 \times 3\)
Breaking down each multiplication step helps in organizing your work and ensuring no terms are skipped over, which makes simplifying equations much easier.
Combining Like Terms
Once you have used the Distributive Property to expand your binomial multiplication, your result will include several terms that can often look a bit messy. This is where combining like terms becomes essential. In our example, after distributing, we get:\(4z^2 + 3z - 8z - 6\)."Like terms" are terms that have the same variable raised to the same power. Hence, \(3z\) and \(-8z\) are like terms because they both contain \(z\). Combine these like terms by simply doing the arithmetic:
  • \(3z - 8z = -5z\)
So, when you combine, it results in:\(4z^2 - 5z - 6\).Looking for and combining like terms reduces the number of terms in your expression, leading to a simpler and easier-to-understand answer.
Polynomial Expansion
Polynomial expansion is the result of applying the Distributive Property to multiply out binomials or polynomials. In our specific example, \((z-2)(4z+3)\), we achieve expansion through systematic distribution and multiplication. By the end of this process, the expanded form of our original expression is \(4z^2 - 5z - 6\).This expanded form is a quadratic polynomial since it includes \(z^2\), z, and constant terms. The expansion transforms an expression that appears compact into one that displays each term distinctly. Mastering this technique is essential for simplifying expressions and solving more complex algebraic equations that involve multiple polynomial terms.

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

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \left(-2 x^{-3} y^{-2}\right)^{3} $$

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \frac{(2 r t)^{2}}{\left(r t^{4}\right)^{-2}} $$

Multiply the polynomials. $$(2-3 x)(5-2 x)\left(x^{2}-1\right)$$

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \frac{\left(d^{3}\right)^{-2}}{\left(d^{-2}\right)^{3}} $$

Multiply the polynomials. $$-x\left(2 x^{4}-x^{2}+10\right)$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.