Chapter 0: Problem 52
Find the product.\(\left(x^{2}+3 x-2\right)\left(x^{2}-3 x-2\right)\)
Short Answer
Expert verified
The product of the given binomial expressions is \(x^{4}+4\)
Step by step solution
01
Distribute the first terms
First, multiply the first terms of each binomial, which are \(x^2\) and \(x^2\). Multiplying these together gives \(x^{2}\cdot x^{2}=x^{4}\)
02
Distribute the Outer terms
Next, multiply the outer terms which are \(x^2\) from the first binomial and \(-3x\) from the second. This gives you \(x^{2}\cdot-3x=-3x^{3}\)
03
Distribute the Inner terms
Then, multiply the inner terms which are \(3x\) from the first binomial and \(x^{2}\) from the second. This gives you \(3x\cdot x^{2}=3x^{3}\)
04
Distribute the last terms
Finally, multiply the last terms of each binomial, which are \(-2\) and \(-2\). This gives you \(-2\cdot -2=4\)
05
Combine like terms
Combine the terms from steps 2 and 3, which are alike because they both involve \(x^{3}\). We have \(-3x^{3}+3x^{3} = 0\). Our final result then is \(x^{4}+4\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributing Terms in Polynomial Multiplication
When multiplying polynomials, one of the key strategies is to distribute each term in the first polynomial to every term in the second polynomial. This process is a foundational part of algebra and is known as the distributive property.
In the exercise provided, we begin by distributing the first term of the first binomial, which is \(x^2\), to every term in the second binomial \(x^{2}-3x-2\). We follow this procedure for each term, essentially unfolding the multiplication into several smaller and more manageable steps. By methodically distributing terms, we ensure that no products are missed and that each term is accounted for.
Once the distribution is complete, we're left with individual products such as \(x^2\cdot x^2\) and \(x^2\cdot (-3x)\), which we then simplify. This breakdown into smaller parts not only makes the calculation easier but also helps in understanding the structure of polynomial equations.
In the exercise provided, we begin by distributing the first term of the first binomial, which is \(x^2\), to every term in the second binomial \(x^{2}-3x-2\). We follow this procedure for each term, essentially unfolding the multiplication into several smaller and more manageable steps. By methodically distributing terms, we ensure that no products are missed and that each term is accounted for.
Once the distribution is complete, we're left with individual products such as \(x^2\cdot x^2\) and \(x^2\cdot (-3x)\), which we then simplify. This breakdown into smaller parts not only makes the calculation easier but also helps in understanding the structure of polynomial equations.
Combining Like Terms to Simplify Expressions
After distributing and multiplying, often there is a need to combine like terms to simplify the polynomial expression. Like terms are those terms that have the same variables raised to the same powers. The coefficients of these terms may be different, but their variable parts are identical.
For example, in the exercise, after performing the distribution, we had \(3x^3\) and \-3x^3\ as products from the inner and outer terms, respectively. These are like terms because they both contain the variable \(x\) raised to the third power. By combining them, the terms cancel each other out, resulting in zero. This simplification is crucial in obtaining a more compact and manageable form of the polynomial, \(x^4+4\). It is vital to check for like terms after multiplication to ensure the expression is fully simplified.
For example, in the exercise, after performing the distribution, we had \(3x^3\) and \-3x^3\ as products from the inner and outer terms, respectively. These are like terms because they both contain the variable \(x\) raised to the third power. By combining them, the terms cancel each other out, resulting in zero. This simplification is crucial in obtaining a more compact and manageable form of the polynomial, \(x^4+4\). It is vital to check for like terms after multiplication to ensure the expression is fully simplified.
Understanding Binomial Products
The exercise introduces the multiplication of two binomials. A binomial is an algebraic expression containing two terms, such as \(a+b\) or \(c-d\). In our example, we multiplied the binomials \(x^2+3x-2\) and \(x^2-3x-2\).
When multiplying binomials, it's important to apply the distributive property to each term, sometimes referred to as the FOIL method (First, Outer, Inner, Last). By multiplying the First terms, Outer terms, Inner terms, and Last terms, we ensure all possible products are captured. The FOIL method provides a systematic approach to binomial multiplication and serves as a useful mnemonic for students learning polynomial algebra.
When multiplying binomials, it's important to apply the distributive property to each term, sometimes referred to as the FOIL method (First, Outer, Inner, Last). By multiplying the First terms, Outer terms, Inner terms, and Last terms, we ensure all possible products are captured. The FOIL method provides a systematic approach to binomial multiplication and serves as a useful mnemonic for students learning polynomial algebra.
Navigating Algebraic Expressions
Algebraic expressions represent a cornerstone concept in algebra and consist of numbers, variables, and arithmetic operations. The complexity of an algebraic expression ranges from simple binomials, like in our exercise, to larger polynomials with multiple terms.
Properly handling algebraic expressions involves being comfortable with variables and exponents, performing arithmetic operations, and understanding how to manipulate these expressions using the distributive property, combining like terms, and other algebraic rules. Mastery of these expressions provides a foundation for solving equations and understanding the relationships between variables within mathematical models.
As you work through problems involving algebraic expressions, ensure you are clear on the rules of exponents, comfortable with the terms you're combining, and precise in your distribution to avoid errors and fully grasp the mathematical concepts at play.
Properly handling algebraic expressions involves being comfortable with variables and exponents, performing arithmetic operations, and understanding how to manipulate these expressions using the distributive property, combining like terms, and other algebraic rules. Mastery of these expressions provides a foundation for solving equations and understanding the relationships between variables within mathematical models.
As you work through problems involving algebraic expressions, ensure you are clear on the rules of exponents, comfortable with the terms you're combining, and precise in your distribution to avoid errors and fully grasp the mathematical concepts at play.
