Problem 11
Question
Add the given polynomials. \(3 x-7\) and \(7 x+4\)
Step-by-Step Solution
Verified Answer
The sum of the polynomials is \(10x - 3\).
1Step 1: Arrange the Polynomials Vertically
To add the polynomials, start by arranging them vertically. Write each polynomial on top of each other, aligning like terms (terms with the same variable and exponent).\[\begin{array}{r}3x -7 \+7x +4\end{array}\]
2Step 2: Add the Coefficients of Like Terms
Add the coefficients of the like terms. For the terms with \(x\), add the coefficients 3 and 7, and for the constant terms, add \(-7\) and 4.\[\begin{array}{r}3x - 7 \+7x + 4 \\hline(3 + 7)x + (-7 + 4)\end{array}\]
3Step 3: Simplify the Result
Perform the calculations to simplify the expression to the simplest form. For the \(x\) terms: \(3 + 7 = 10\) so we get 10x.For the constant terms: \(-7 + 4 = -3\).The result is:\[10x - 3\]
4Step 4: Write the Final Answer
Combine the simplified terms into a single polynomial expression:\[10x - 3\] The sum of the polynomials \(3x - 7\) and \(7x + 4\) is \(10x - 3\).
Key Concepts
Polynomial OperationsAlgebraic ExpressionsLike Terms
Polynomial Operations
Polynomial operations, such as addition, are fundamental skills in algebra. Adding polynomials might seem like a small task, but it’s crucial for solving more complex algebraic problems.
When adding polynomials, the key is to focus on combining like terms. This means looking for terms that have the same variable raised to the same power, like two terms both involving \(x\). All polynomials consist of terms that can include numbers (called coefficients), variables (such as \(x\) or \(y\)), and exponents. By performing basic operations on polynomials, such as addition, you can simplify expressions and solve equations more efficiently.
Be sure to arrange the polynomials correctly, usually vertically, as shown in the solution. This makes it easier to visually match and add the like terms. Also, remember that these operations work in a linear way, so only like terms can be added or subtracted directly without any multiplication or division involved.
Understanding these operations lays the groundwork for future topics such as polynomial multiplication, division, and factorization.
When adding polynomials, the key is to focus on combining like terms. This means looking for terms that have the same variable raised to the same power, like two terms both involving \(x\). All polynomials consist of terms that can include numbers (called coefficients), variables (such as \(x\) or \(y\)), and exponents. By performing basic operations on polynomials, such as addition, you can simplify expressions and solve equations more efficiently.
Be sure to arrange the polynomials correctly, usually vertically, as shown in the solution. This makes it easier to visually match and add the like terms. Also, remember that these operations work in a linear way, so only like terms can be added or subtracted directly without any multiplication or division involved.
Understanding these operations lays the groundwork for future topics such as polynomial multiplication, division, and factorization.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (like addition, subtraction, multiplication, and division). These expressions allow us to express mathematical relationships or real-world problems in a concise form.
Polynomials are a type of algebraic expression composed of one or more terms. In the given exercise, each polynomial consists of two terms: a term involving \(x\), like \(3x\), and a constant term, such as \(-7\). Unlike equations, algebraic expressions don’t have an equality sign; they simply represent a value or an expression.
When working with algebraic expressions, particularly polynomials, the idea is to manipulate them to achieve a desired form or solution. This might involve simplifying the expression, combining like terms, or finding an equivalent expression. These manipulations are the basis for many algebra problems and solutions.
Grasping how algebraic expressions operate helps you comfortably tackle more advanced mathematical concepts, be it in calculus or even physics.
Polynomials are a type of algebraic expression composed of one or more terms. In the given exercise, each polynomial consists of two terms: a term involving \(x\), like \(3x\), and a constant term, such as \(-7\). Unlike equations, algebraic expressions don’t have an equality sign; they simply represent a value or an expression.
When working with algebraic expressions, particularly polynomials, the idea is to manipulate them to achieve a desired form or solution. This might involve simplifying the expression, combining like terms, or finding an equivalent expression. These manipulations are the basis for many algebra problems and solutions.
Grasping how algebraic expressions operate helps you comfortably tackle more advanced mathematical concepts, be it in calculus or even physics.
Like Terms
Like terms in algebra are terms that share the same variable with the same exponent. They are critical in simplifying and performing operations on algebraic expressions. In the exercise, the like terms are those that involve \(x\) – like \(3x\) and \(7x\) – as well as the constant terms \(-7\) and \(4\).
Combining like terms is simple: add or subtract the coefficients (the numbers in front of the variables). For example, in our exercise, \(3x + 7x\) becomes \(10x\), and then \(-7 + 4\) simplifies to \(-3\). These calculations lead to a simplified expression with fewer terms, making it easier to understand and work with.
Recognizing and combining like terms reduces complexity in expressions and equations, giving you a cleaner and more manageable representation of a problem. This concept is used frequently in algebra, and mastering it is critical for success in math.
Combining like terms is simple: add or subtract the coefficients (the numbers in front of the variables). For example, in our exercise, \(3x + 7x\) becomes \(10x\), and then \(-7 + 4\) simplifies to \(-3\). These calculations lead to a simplified expression with fewer terms, making it easier to understand and work with.
Recognizing and combining like terms reduces complexity in expressions and equations, giving you a cleaner and more manageable representation of a problem. This concept is used frequently in algebra, and mastering it is critical for success in math.
Other exercises in this chapter
Problem 11
Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(a-3 b)(c+4 d)$$
View solution Problem 11
Find each product. $$\left(3 a^{2} b\right)\left(9 a^{2} b^{4}\right)$$
View solution Problem 12
Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$n(n-24)=-128$$
View solution Problem 12
Factor completely each of the polynomials and indicate any that are not factorable using integers. $$35-2 x-x^{2}$$
View solution