Problem 1
Question
Fill in the blanks. \(x^{2}+6 x+9\) is a _________ -square trinomial because it is the square of the binomial \(x+3\).
Step-by-Step Solution
Verified Answer
Perfect
1Step 1: Identify the general form of a perfect square trinomial
A perfect square trinomial is in the form \( (a+b)^2 \) which expands to \( a^2 + 2ab + b^2 \). Our goal is to identify if the given trinomial \( x^2 + 6x + 9 \) fits this format.
2Step 2: Compare the given trinomial with the perfect square form
The given trinomial is \( x^2 + 6x + 9 \). We compare it with the general form \( a^2 + 2ab + b^2 \). Here, \( a^2 \) corresponds to \( x^2 \), giving \( a = x \), and \( b^2 \) corresponds to \( 9 \), giving \( b = 3 \). Also, check if the middle term \( 6x \) corresponds to \( 2ab \), which is \( 2 \times x \times 3 = 6x \). This confirms the fit.
3Step 3: Confirm if the trinomial is a perfect square
Since all terms match the pattern of a perfect square trinomial, \( x^2 + 6x + 9 = (x+3)^2 \) is indeed a perfect square trinomial.
Key Concepts
Understanding Algebra BasicsExploring Trinomial FactoringThe Role of Polynomials in Mathematics
Understanding Algebra Basics
Algebra serves as the foundation of many mathematical concepts. At its core, algebra involves the use of symbols, often letters, to represent numbers in equations and formulas. This allows us to create relationships between different quantities and find solutions systematically.
In algebraic expressions, each term is a component of the expression, usually consisting of a coefficient, a variable, and an exponent. For example, in the term \(6x\), 6 is the coefficient, \(x\) is the variable, and the exponent is 1 since it’s not visibly shown.
When working with algebra, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Correctly applying this order ensures that we solve equations accurately.
In algebraic expressions, each term is a component of the expression, usually consisting of a coefficient, a variable, and an exponent. For example, in the term \(6x\), 6 is the coefficient, \(x\) is the variable, and the exponent is 1 since it’s not visibly shown.
When working with algebra, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Correctly applying this order ensures that we solve equations accurately.
Exploring Trinomial Factoring
Trinomial factoring is a specific technique in algebra used to simplify polynomials, specifically those with three terms. It is essential when solving equations or simplifying expressions. The process involves finding two binomial expressions that multiply together to give the original trinomial.
A perfect square trinomial, like \(x^2 + 6x + 9\), can be factored into \((x + 3)^2\). This type of trinomial comes from expanding a squared binomial, which fits the general form \((a + b)^2 = a^2 + 2ab + b^2\).
To factor a trinomial:
A perfect square trinomial, like \(x^2 + 6x + 9\), can be factored into \((x + 3)^2\). This type of trinomial comes from expanding a squared binomial, which fits the general form \((a + b)^2 = a^2 + 2ab + b^2\).
To factor a trinomial:
- Identify \(a\) and \(b\) in the format \(a^2 + 2ab + b^2\) from the given trinomial.
- Ensure that the middle term matches \(2ab\).
- Write the factored form as \((a + b)^2\).
The Role of Polynomials in Mathematics
Polynomials are algebraic expressions made up of multiple terms combined through addition, subtraction, and multiplication. They are very versatile and appear in a variety of mathematical contexts, from basic algebra to calculus, and even in real-world applications.
Most commonly, polynomials are classified by the number of terms they have. A trinomial, for example, is a type of polynomial with three terms, like \(x^2 + 6x + 9\). Another important classification is by degree, which is determined by the highest exponent of the variable present in the expression.
Polynomials are foundational in algebra and other branches of mathematics because they represent both simple and complex relationships between numbers. They can often be solved using factoring techniques—such as the factorization of trinomials—which simplify expressions and equations, making solving them more straightforward.
Most commonly, polynomials are classified by the number of terms they have. A trinomial, for example, is a type of polynomial with three terms, like \(x^2 + 6x + 9\). Another important classification is by degree, which is determined by the highest exponent of the variable present in the expression.
Polynomials are foundational in algebra and other branches of mathematics because they represent both simple and complex relationships between numbers. They can often be solved using factoring techniques—such as the factorization of trinomials—which simplify expressions and equations, making solving them more straightforward.
Other exercises in this chapter
Problem 1
Integers that follow one another, such as 6 and \(7,\) are called ___________ integers.
View solution Problem 1
Fill in the blanks. \(2 x^{2}+3 x-1=0\) and \(x^{2}-36=0\) are examples of ___equations.
View solution Problem 1
\(x^{3}+27\) is the ____ of two cubes and \(a^{3}-125\) is the difference of two ____.
View solution Problem 1
Fill in the blanks. The trinomial \(x^{2}-x-12 _____ as the product of two binomials: \)(x-4)(x+3)$
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