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Chapter 9: Problem 11

Subtract the following polynomials. $$\left(4 x^{2}+3 x+2\right)-\left(2 x^{2}+5 x+1\right)$$

Short Answer

Expert verified
The difference of the polynomials is \(2x^2 - 2x + 1\).

Step by step solution

01

Write down the polynomials

First, clearly write the polynomials we are going to subtract: The first polynomial is \(4x^2 + 3x + 2\).The second polynomial is \(2x^2 + 5x + 1\).

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Key Concepts

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

Polynomials
A polynomial is an algebraic expression consisting primarily of variables and coefficients. The expression is made up of terms, which can include a constant (a standalone number), a variable (like \(x\)), and an exponent (showing the power to which the variable is raised).
For example, in the expression \(4x^2 + 3x + 2\),
  • \(4x^2\) is a term where 4 is the coefficient, \(x\) is the variable, and 2 is the exponent.
  • \(3x\) is another term with 3 as the coefficient and the exponent implied to be 1.
  • The 2 is a constant term with no variable attached.
Polynomials can be classified by degree, which is the largest exponent of a variable in the polynomial. In the given polynomial \(4x^2 + 3x + 2\), the degree is 2, as the highest power of \(x\) is 2.
Like Terms
Like terms are terms that contain the same variables raised to the same power. This means that only the coefficients are different. It's important to identify like terms when combining or subtracting terms in polynomials.

In the example \(4x^2 + 3x + 2\) and \(2x^2 + 5x + 1\),
  • \(4x^2\) and \(2x^2\) are like terms because they contain the same variable \(x\) raised to the power of 2.
  • \(3x\) and \(5x\) are like terms because they both contain \(x\) to the power of 1.
  • The constants \(2\) and \(1\) are also like terms because they are standalone numbers with no variable.
Recognizing like terms is crucial for simplifying polynomials, as only like terms can be combined through addition or subtraction.
Distributive Property
The distributive property is a fundamental concept in algebra that allows you to multiply a sum by multiplying each addend separately and then adding the products. It is typically expressed as \(a(b + c) = ab + ac\).
However, when subtracting polynomials, you can think of the distributive property as distributing the negative sign across the terms of the second polynomial.
  • Apply the negative sign from the subtraction to each term in the polynomial being subtracted.
  • In our exercise: turning \(\left(2x^2 + 5x + 1\) into \((-2x^2 - 5x - 1)\).
  • This helps ensure accuracy in subtraction, where sign errors are common.
By distributing the negative sign correctly, you prevent mistakes and simplify the subtraction process.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations (like addition or subtraction). Unlike equations, they do not involve equality signs. They serve as the foundation for forming problems in algebra.

Algebraic expressions can be simple, like \(x + 2\), or more complex, such as \(4x^2 + 3x + 2\). When working with algebraic expressions, you:
  • Combine like terms when possible to simplify the expression.
  • Use arithmetic operations to rearrange or solve problems as needed.
Understanding algebraic expressions is pivotal in solving more complex algebraic equations and inequalities, as they allow you to model real-world scenarios mathematically.

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Most popular questions from this chapter

Multiply the following expressions. \(\left(-2 x^{5} y^{2}\right)\left(-4 x^{3} y^{4}\right)\)

Multiply the following expressions. $$4 x^{2} y^{3}\left(3 x^{4}+2 y^{3}\right)$$

Find each quotient. Write all answers in scientific notation. $$\frac{0.00092}{46,000}$$

Multiply the following expressions. \(\left(8 a^{4} b^{4}\right)\left(5 a^{3} b^{5}\right)\left(a^{2} b^{6}\right)\)

Simplify the following expressions. \(\left(5 x^{2} y^{3}\right)^{4}\)
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