Problem 86
Question
For the following problems, perform the multiplications and combine any like terms. $$ \left(x^{2}+5\right)(x+4) $$
Step-by-Step Solution
Verified Answer
Question: Multiply the expressions \((x^2 + 5)\) and \((x + 4)\) and find the final expression.
Answer: The final expression after multiplication is \(x^3 + 4x^2 + 5x + 20\).
1Step 1: Distribute the first term of the first expression
Multiply \(x^2\) by both terms in the second expression:
$$
x^2 \cdot (x + 4) = x^2 \cdot x + x^2 \cdot 4 \\
= x^3 + 4x^2
$$
2Step 2: Distribute the second term of the first expression
Multiply 5 by both terms in the second expression:
$$
5 \cdot (x+4) = 5 \cdot x + 5 \cdot 4 \\
= 5x + 20
$$
3Step 3: Combine the results from Steps 1 and 2
Add the results from Steps 1 and 2 to get the final expression:
$$
(x^3 + 4x^2) + (5x + 20) \\
= x^3 + 4x^2 + 5x + 20
$$
So, after performing the multiplication and combining like terms, the final expression is:
$$
x^3 + 4x^2 + 5x + 20
$$
Key Concepts
Combining Like TermsDistributive PropertyAlgebraic Expressions
Combining Like Terms
Understanding how to combine like terms is essential when simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. In the exercise \(x^3 + 4x^2 + 5x + 20\), for instance, there are no like terms to combine since each term is different in terms of the degree of \(x\).
However, recognizing like terms in more complex expressions is a critical skill. For example, if you had an expression like \(3x^2 + 2x + 5 + 4x^2 + x\), you would combine the terms with \(x^2\) to get \(7x^2\), and the terms with \(x\) to get \(3x\), resulting in \(7x^2 + 3x + 5\). The process of combining like terms typically follows addition or multiplication operations, streamlining the expression to make it more manageable.
However, recognizing like terms in more complex expressions is a critical skill. For example, if you had an expression like \(3x^2 + 2x + 5 + 4x^2 + x\), you would combine the terms with \(x^2\) to get \(7x^2\), and the terms with \(x\) to get \(3x\), resulting in \(7x^2 + 3x + 5\). The process of combining like terms typically follows addition or multiplication operations, streamlining the expression to make it more manageable.
Distributive Property
The distributive property, sometimes known as the distributive law of multiplication and division, is one of the most frequently used properties in algebra. It allows you to multiply a single term across terms inside a parenthesis. The general formula is \(a(b + c) = ab + ac\).
In the provided exercise, this property is applied in two steps. First, \(x^2\) is distributed over \(x + 4\), leading to \(x^2 \cdot x + x^2 \cdot 4\); then \(5\) is distributed over the same binomial, giving us \(5x + 20\). The application of the distributive property helps break down the expression into simpler parts, which can then be added or combined in the subsequent steps to reach the simplified form of the polynomial.
In the provided exercise, this property is applied in two steps. First, \(x^2\) is distributed over \(x + 4\), leading to \(x^2 \cdot x + x^2 \cdot 4\); then \(5\) is distributed over the same binomial, giving us \(5x + 20\). The application of the distributive property helps break down the expression into simpler parts, which can then be added or combined in the subsequent steps to reach the simplified form of the polynomial.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and arithmetic operations (addition, subtraction, multiplication, division, and exponentiation). In the context of our exercise \( (x^{2}+5)(x+4) \), the expression is a product of two binomials, which is a common type of algebraic expression.
When multiplying polynomials, especially binomials, each term in the first polynomial is multiplied by each term in the second polynomial, as was broken down in the step-by-step solution. Mastery of reading and constructing algebraic expressions is key to understanding and solving a wide variety of problems in algebra. Remember, expressions don't have an equals sign; when they do, they become equations. Algebraic expressions, like the ones you interact with in this exercise, are the building blocks for more complex mathematical concepts.
When multiplying polynomials, especially binomials, each term in the first polynomial is multiplied by each term in the second polynomial, as was broken down in the step-by-step solution. Mastery of reading and constructing algebraic expressions is key to understanding and solving a wide variety of problems in algebra. Remember, expressions don't have an equals sign; when they do, they become equations. Algebraic expressions, like the ones you interact with in this exercise, are the building blocks for more complex mathematical concepts.
Other exercises in this chapter
Problem 85
Simplify the algebraic expressions for the following problems. $$ \text { Add }-3 x+4 \text { to } 5 x-8 \text { . } $$
View solution Problem 85
For the following problems, simplify each of the algebraic expressions. $$ -4(2 x-3 y) $$
View solution Problem 86
Simplify the algebraic expressions for the following problems. $$ \text { Add } 4\left(x^{2}-2 x-3\right) \text { to }-6\left(x^{2}-5\right) $$
View solution Problem 86
For the following problems, simplify each of the algebraic expressions. $$ -4 x y^{2}\left[7 x y-6\left(5-x y^{2}\right)+3(-x y+1)+1\right] $$
View solution