Problem 122 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The polynomial \(16 x^{2}+20 x+25\) is a perfect square trinomial.

Step-by-Step Solution

Answer

The statement is False, the correct perfect square trinomial would be \(16x^2+40x+25\).

Step 1: Identify the values of 'a', 'b', and 'c'

For the quadratic polynomial in the form of \(ax^2 + bx + c\), we obtain the values as a\(=\sqrt{16}=4\), b=\(\frac{20}{2}=10\), and c\(=\sqrt{25}=5\)

Step 2: Check for perfect square trinomial

A trinomial \((a^2+2ab+b^2)\) is considered a perfect square trinomial if \(2ab = b\). This means, we need to check whether or not \(2*a*c == b\). In our case, calculate \(2*4*5 == 10\). This calculation gives the result of '40', which is not equal to 'b'. Therefore, our given polynomial is not in the perfect square trinomial form.

Step 3: Modify the polynomial to make a correct statement

From the previous step, we understand that our given polynomial is not a perfect square trinomial. Now according to the problem statement, we should make necessary changes to produce a true statement. So, the statement will be true if the polynomial is \(16x^2+40x+25\) instead of \(16x^2+20x+25\)

Key Concepts

Polynomial Identities

Polynomial identities are equations that hold true for all values of the variables involved. They serve as useful tools in algebra for simplifying expressions and solving problems. An important type of polynomial identity is the perfect square trinomial. When you encounter expressions like

• \(a^2 + 2ab + b^2 = (a + b)^2\)

They represent a form of polynomial identity where a trinomial is expressed as the square of a binomial.

In algebraic manipulation, recognizing these identities helps you to factor expressions, simplify equations, and solve algebraic problems easily. Some common examples include:

• Sum of squares: \((a+b)^2 = a^2 + 2ab + b^2\)
• Difference of squares: \((a-b)^2 = a^2 - 2ab + b^2\)

These identities are integral for identifying or transforming expressions into recognizable forms, bringing accuracy and simplicity into algebraic processes.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two, typically represented in the standard form: \(ax^2 + bx + c\). Here:

• \(a\), \(b\), and \(c\) are constants.
• \(a\) cannot be zero, as this would otherwise represent a linear polynomial.
• \(x\) denotes the variable.

Quadratic polynomials appear frequently in algebra and are fundamental to the study of polynomial equations. They can be solved using different methods such as factoring, completing the square, and using the quadratic formula

In the context of perfect square trinomials, you might come across such quadratic polynomials that can be rewritten as the square of a binomial.

Key characteristics include:

• The parabola shape when plotted on a graph.
• Their ability to model a wide range of real-world phenomena.

Recognizing a quadratic polynomial helps in solving complex problems and understanding its practical applications.

Algebra Concepts

Algebra encompasses various concepts that enable us to handle numbers and variables making it essential for solving equations and understanding mathematical relationships. Among the core topics in algebra is the identification and manipulation of polynomial identities and quadratic polynomials.

Grasping these concepts can be simplified by understanding:

• The structure and characteristics of polynomials and how they factor into algebraic expressions.
• The importance of operations such as addition, subtraction, multiplication, and division of polynomials.
• Special cases like perfect square trinomials that signify powerful shortcuts in dealing with polynomials and equations.

By mastering algebra concepts, students gain a robust mathematical foundation, making complex subjects more approachable. It also aids in developing problem-solving skills that apply in various fields beyond mathematics, such as physics, engineering, and economics.