Chapter 2: Problem 6
Multiply and simplify.$$\left(a^{2}-3 b\right)\left(a^{2}+5 b\right)$$
Short Answer
Expert verified
\(a^4 + 2a^2b - 15b^2\)
Step by step solution
01
Apply the FOIL Method
To multiply the two binomials, use the FOIL (First, Outer, Inner, Last) method. This means you multiply the first terms of each binomial together, then the outer terms, followed by the inner terms, and finally the last terms of each binomial.
02
Multiply the First Terms
Multiply the first terms of each binomial: \(a^2 \times a^2 = a^{2+2} = a^4\).
03
Multiply the Outer Terms
Multiply the outer terms of each binomial: \(a^2 \times 5b = 5a^2b\).
04
Multiply the Inner Terms
Multiply the inner terms of each binomial: \(-3b \times a^2 = -3a^2b\).
05
Multiply the Last Terms
Multiply the last terms of each binomial: \(-3b \times 5b = -15b^2\).
06
Combine Like Terms
Combine like terms from the multiplication: \(a^4 + 5a^2b - 3a^2b - 15b^2\). Note that \(5a^2b - 3a^2b = 2a^2b\).
07
Write the Final Simplified Expression
After combining like terms, the final simplified expression is \(a^4 + 2a^2b - 15b^2\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Multiplication of Binomials
The multiplication of binomials is a common task in algebra that is made easy with the FOIL method. The term 'binomial' refers to a polynomial with two terms, which are typically written in the form \(x + y\) or \(x - y\). When we need to multiply two binomials together, we follow the FOIL process, which stands for First, Outer, Inner, Last. This methodically multiplies each term in the first binomial by each term in the second binomial.
For instance, when multiplying the binomials \(a^2 - 3b\) and \(a^2 + 5b\), we multiply the first terms \(a^2\) and \(a^2\), the outer terms \(a^2\) and \(5b\), the inner terms \(3b\) and \(a^2\), and finally the last terms \(3b\) and \(5b\). This systematic approach ensures that all possible products are computed, which is the first critical step in simplifying the expression.
For instance, when multiplying the binomials \(a^2 - 3b\) and \(a^2 + 5b\), we multiply the first terms \(a^2\) and \(a^2\), the outer terms \(a^2\) and \(5b\), the inner terms \(3b\) and \(a^2\), and finally the last terms \(3b\) and \(5b\). This systematic approach ensures that all possible products are computed, which is the first critical step in simplifying the expression.
Combining Like Terms
Once the individual products of a binomial multiplication have been found using the FOIL method, the next step involves combining like terms. Like terms are terms that have the same variables raised to the same power. They are the building blocks of algebraic simplification, as they allow us to compact expressions into more manageable forms.
In our example with the expression \(a^4 + 5a^2b - 3a^2b - 15b^2\), we identify \(5a^2b\) and \(3a^2b\) as like terms because they share the same variables with identical exponents. By combining these like terms, which involves adding their coefficients—5 and -3—we simplify the expression to \(a^4 + 2a^2b - 15b^2\). This step is vital for reducing complexity and attaining the most simplified version of the expression.
In our example with the expression \(a^4 + 5a^2b - 3a^2b - 15b^2\), we identify \(5a^2b\) and \(3a^2b\) as like terms because they share the same variables with identical exponents. By combining these like terms, which involves adding their coefficients—5 and -3—we simplify the expression to \(a^4 + 2a^2b - 15b^2\). This step is vital for reducing complexity and attaining the most simplified version of the expression.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves several techniques, including the distribution of terms, combining like terms, and applying exponent rules. The goal is to rewrite an expression in its simplest form, where no further arithmetic operations can be performed on the terms. Simplification makes it easier to understand, compare, and use algebraic expressions in further calculations.
After using the FOIL method and combining like terms as demonstrated in our example \(a^4 + 2a^2b - 15b^2\), the expression is considered simplified. It is important to look for any opportunity to reduce terms further through factorization or by applying other algebraic identities. However, in this case, there is no further simplification possible, indicating that the expression is fully simplified.
After using the FOIL method and combining like terms as demonstrated in our example \(a^4 + 2a^2b - 15b^2\), the expression is considered simplified. It is important to look for any opportunity to reduce terms further through factorization or by applying other algebraic identities. However, in this case, there is no further simplification possible, indicating that the expression is fully simplified.
Exponent Rules
Exponent rules, also known as laws of exponents, play a pivotal role in manipulating algebraic expressions, especially when performing operations like multiplication or division on terms with powers. Key rules include the product rule \(a^m \times a^n = a^{m+n}\), the power rule \( (a^m)^n = a^{m \times n}\), and others that deal with quotients and zero exponents.
These rules are perfectly illustrated in the multiplication of our first terms from the provided binomials: \(a^2 \times a^2 = a^{2+2} = a^4\). By adding the exponents, we effectively apply the product rule of exponents. A solid understanding of these exponent rules is essential as they are frequently used in algebra to simplify and solve expressions and equations.
These rules are perfectly illustrated in the multiplication of our first terms from the provided binomials: \(a^2 \times a^2 = a^{2+2} = a^4\). By adding the exponents, we effectively apply the product rule of exponents. A solid understanding of these exponent rules is essential as they are frequently used in algebra to simplify and solve expressions and equations.
