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Chapter 4: Problem 33

Multiply. $$-4(t-7)$$

Short Answer

Expert verified
The solution is \(-4t + 28\).

Step by step solution

01

Identify the elements to be multiplied

The equation has a number -4 outside the parentheses which needs to be multiplied by each individual item within parentheses, \(t\) and \(-7\).
02

Multiply -4 with each term inside the parentheses

Multiply -4 with \(t\) to get \(-4t\), and -4 with \(-7\) to get \(28\).
03

Write the final expression

The final expression after multiplication is \(-4t + 28\).

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Key Concepts

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

Multiplication in Algebra
Multiplication is a crucial operation in algebra that involves repeated addition of a number. In algebra, you often use multiplication to simplify expressions or solve equations. For example, if you have a term like \(-4(t-7)\), multiplication is used to distribute \(-4\) to both terms inside the parenthesis. The purpose is to remove the parenthesis and combine individual terms into a simpler expression. Here's why this is important:
  • It helps clear complex algebraic expressions making them easier to solve.
  • It's a step towards balancing equations and finding variable values.
So, understanding multiplication beyond simple arithmetic is essential to excel in algebra.
Understanding the Distributive Property
The distributive property is an algebraic principle used to break down expressions involving multiplication across terms inside parentheses. It tells us how to distribute the multiplication of a number with each term inside the parenthesis. For example, with the expression \(-4(t-7)\), you apply this property by multiplying \(-4\) with each term: first \(t\) and then \(-7\).This property is invaluable for simplifying expressions because:
  • It helps in dealing with expressions of unusual form.
  • Ensures each term is accounted for, preventing calculation errors.
Grasping this concept allows you to effortlessly transform and solve complex problems.
What is a Polynomial?
Polynomials are expressions made up of variables and coefficients, involving only addition, subtraction, and multiplication. They include terms like \(-4t + 28\), which are combinations of constants and variables. Polynomials can vary in complexity:
  • Single-term: like \(-4t\).
  • Multi-term: like \(-4t + 28\), known as a binomial (two terms).
  • More than two terms are trinomials or higher-degree polynomials.
Understanding polynomials is vital because they form the basis of many algebraic equations. They allow you to work with expressions that model real-world data and solve various practical problems.

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Most popular questions from this chapter

Simplify or solve as appropriate. $$3 y(y+2)=3(y+1)(y-1)$$

Perform the operations. $$\begin{array}{r}-3 x^{2} y+4 x y+25 y^{2} \\\\-\quad 5 x^{2} y-3 x y-12 y^{2}\end{array}$$

Perform each division. If there is a remainder, leave the answer in quotient \(+\frac{\text { remainder }}{\text { divisor }}\) form. Assume no division by \(0 .\) $$\frac{2 x^{3}+4 x^{2}-2 x+3}{x-2}$$

How would you check the results of a division?

$$\frac{x^{3}+y^{3}}{x+y}$$
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