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Chapter 1: Problem 9

Express as a polynomial. $$ (2 u+3)(u-4)+4 u(u-2) $$

Short Answer

Expert verified
The polynomial is \(6u^2 - 13u - 12\).

Step by step solution

01

Expand the First Product

First, we need to expand the expression \((2u + 3)(u - 4)\). This gives us:\[ 2u(u - 4) + 3(u - 4) = 2u^2 - 8u + 3u - 12 = 2u^2 - 5u - 12. \]
02

Expand the Second Term

Next, expand the expression \(4u(u - 2)\) as follows:\[ 4u^2 - 8u.\]
03

Combine Like Terms

Now, add the expressions obtained from Step 1 and Step 2:\[ (2u^2 - 5u - 12) + (4u^2 - 8u). \]Combine like terms:- Combine the \(u^2\) terms: \(2u^2 + 4u^2 = 6u^2\).- Combine the \(u\) terms: \(-5u - 8u = -13u\).- The constant term remains: \(-12\).This results in the final polynomial:\[ 6u^2 - 13u - 12.\]

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Key Concepts

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

Combining Like Terms
Combining like terms is a fundamental concept in algebra. It allows you to simplify expressions by gathering terms that have the same variable raised to the same power. This process helps in simplifying expressions for ease of further manipulations.

In a polynomial, you identify terms with the same variable parts, known as like terms.
  • If the terms have the same variable and exponent, you simply add or subtract their coefficients.
  • This means that you're dealing only with the numbers in front of the variables, known as coefficients, making the process straightforward.
In our exercise, we have terms like:
  • \(2u^2\) and \(4u^2\) are like terms because both are squared terms involving the variable \(u\).
  • Similarly, \(-5u\) and \(-8u\) are like terms because both are linear terms with \(u\).
By adding these like terms together, we achieve a more simplified and manageable polynomial expression.
Binomial Multiplication
Binomial multiplication is the process of multiplying two binomials. A binomial is an algebraic expression containing two terms, such as \((a + b)(c + d)\).

To multiply binomials, you can use the distributive property:
  • Each term in the first binomial multiplies with each term in the second.
  • This results in four products, which you then need to sum.
For our exercise,
  • First, multiply \((2u + 3)\) by \((u - 4)\), giving terms like \(2u(u)\), \(2u(-4)\), \(3(u)\), and \(3(-4)\).
  • This results in the expression \(2u^2 - 8u + 3u - 12\).
  • Then, multiply \(4u\) by \((u - 2)\) to get the terms \(4u^2\) and \(-8u\).
It's essential to carefully organize and simplify these expressions to minimize errors.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Unlike a full equation, an expression does not have an equality sign.

These expressions often contain terms made up of coefficients (numbers in front of variables), variables, and constants (stand-alone numbers).
  • In the given problem, expressions like \((2u + 3)(u - 4)\) and \(4u(u - 2)\) showcase the combination of multiple elements.
  • Understanding these components helps in systematically solving and simplifying expressions.
Evaluating algebraic expressions involves substituting the variables with numerical values or simplifying them through combining like terms and multiplication techniques. By mastering these skills, you gain a deeper insight into how algebra works as a whole.

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Most popular questions from this chapter

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{2+9 i}{-3-i} $$

Factor the polynomial. $$ 81 r^{2}-16 t^{2} $$

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (2+5 i)^{3} $$

Find the values of \(x\) and \(y\), where \(x\) and \(y\) are $$ 4+(x+2 y) i=x+2 i $$

Factor the polynomial. $$ 75 x^{2}-48 y^{2} $$
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