Problem 97 - Algebra A Combined Function

Question

Fill in the blank so that \(x^{2}+_______x+16\) is a perfect square trinomial.

Step-by-Step Solution

Answer

The blank should be filled with 8.

Step 1: Understanding Perfect Square Trinomials

A perfect square trinomial is in the form of \(a^2 + 2ab + b^2\) and can be expressed as \((a+b)^2\). This means that we'll need to identify what values correspond to \(2ab\) and \(b^2\). In the trinomial provided, \(x^2 + _______x + 16\), we recognize that \(16\) is \(b^2\), meaning \(b = 4\) because \(4^2 = 16\).

Step 2: Calculating the Missing Coefficient

Now identify \(2ab\). Given that \(b = 4\), we need to find the coefficient for \(x\) in \((a+4)^2 = a^2 + 2 \cdot a \cdot 4 + 16\). Hence, the coefficient of \(x\) is \(2 \times 4 = 8\). Therefore, \(a = 8/2 = 4\). The coefficient of \( x\) should thus be \8\.

Step 3: Verifying the Perfect Square Trinomial

Insert the calculated coefficient into the trinomial: \(x^2 + 8x + 16\). Verify it is a perfect square by expanding \((x+4)^2\). We get \(x^2 + 2\cdot 4x + 4^2 = x^2 + 8x + 16\), which confirms our solution.

Key Concepts

Polynomials

Polynomials are algebraic expressions that involve variables raised to whole number powers and have coefficients as constants. They play an essential role in algebra and are composed of terms combined through addition or subtraction. Each term in a polynomial is made up of:

• Coefficient: The numerical factor in front of a variable, e.g., in 3x, 3 is the coefficient.
• Variable: Often denoted by letters like x or y, which can change and take on different values.
• Exponent: The power to which the variable is raised. In x², 2 is the exponent.

Polynomials can be classified based on their degree:

• Linear Polynomial: The highest degree is one, e.g., 2x + 3.
• Quadratic Polynomial: The highest degree is two, e.g., x² + 3x + 2.
• Cubic Polynomial: The highest degree is three, e.g., 2x³ + x² - x + 5.

Understanding polynomials is crucial because they form the basis for more complex algebraic operations.

Factoring

Factoring is the process of breaking down an algebraic expression into products of simpler expressions or numbers. For polynomials, factoring helps simplify expressions or solve polynomial equations. In the context of perfect square trinomials, factoring allows us to express the trinomial as the square of a binomial.For example, the trinomial \(x^2 + 8x + 16\) can be factored into \((x+4)^2\). This is derived from the identity:\[(a + b)^2 = a^2 + 2ab + b^2\]Here’s how the process generally works:

• Identify 'a' and 'b' such that the constant term is \(b^2\), in this instance, \(16\) (since \(b = 4\)).
• Ensure that the middle term represents \(2ab\). Here, \(2 \times 4 \times 1 = 8\), confirming the coefficient of x is correct.
• Rewrite the polynomial as the square of a binomial once "a" and "b" are found.

This method highlights how factoring simplifies expressions and verifies solutions by reconstructing the original polynomial.

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers, brought together through arithmetic operations like addition, subtraction, multiplication, and division. These expressions form the building blocks of algebra, enabling mathematicians to generalize arithmetic.Key components include:

• Variables: Represent unknown values and are typically denoted by letters such as x, y, or z.
• Constants: Fixed numerical values, like the number 16 in the trinomial given.
• Operators: Symbols like +, -, *, and / that tell you how to combine the values.

An essential property of algebraic expressions is that they can be simplified or transformed, allowing for easier computation or factorization. In our perfect square trinomial exercise, the algebraic expression \(x^2 + 8x + 16\) was constructed to be factored conveniently into a perfect square form.The versatility of algebraic expressions is what makes them fundamental, as they can represent real-world problems or mathematical scenarios with precision.