Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 16 z^{2}-56 z+49 $$

Step-by-Step Solution

Answer

The factored form is \((4z - 7)^2\).

Step 1: Identify the Form of the Polynomial

The given polynomial is a quadratic in the form of \( az^2 + bz + c \), where \( a = 16 \), \( b = -56 \), and \( c = 49 \).

Step 2: Check for Perfect Square Trinomials

We recognize that the quadratic might be a perfect square trinomial. A perfect square trinomial takes the form \((mz + n)^2\) where \( mz \) is the square root of \( az^2 \) and \( n \) is the square root of \( c \).

Step 3: Find the Square Roots

The square root of \( 16z^2 \) is \( 4z \), and the square root of \( 49 \) is \( 7 \). We hypothesize that the polynomial is \((4z - 7)^2\).

Step 4: Expand the Hypothesized Perfect Square

Expand \((4z - 7)^2\): \[(4z - 7)^2 = (4z - 7)(4z - 7) = 16z^2 - 28z - 28z + 49 = 16z^2 - 56z + 49\].The expanded form matches the given polynomial.

Step 5: Verify the Factored Form

Since the expansion results in the original polynomial, \( 16z^2 - 56z + 49 \) is indeed the perfect square of \( (4z - 7) \). Therefore, the factorization is correct.

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that are characterized by the highest degree of the variable being two. This means they take the general form of \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The process of solving quadratic equations can involve finding the roots, or solutions, which are the values of the variable that satisfy the equation. These roots can often be real or complex numbers.

Common methods to solve quadratic equations include:

• Factoring
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square

Understanding quadratic equations is fundamental to progressing in algebra and is applicable in various fields such as physics, engineering, and economics.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that is created by squaring a binomial expression. It takes the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\).

Recognizing perfect square trinomials can simplify factoring significantly because they have a consistent pattern. For example, if you have a trinomial like \( 16z^2 - 56z + 49 \), you can check for perfect squares:

• The first term \( 16z^2 \) is \((4z)^2\).
• The last term \( 49 \) is \(7^2\).
• The middle term \(-56z\) can be checked by calculating \(2 \times 4z \times 7 = -56z\), which matches.

In this way, perfect square trinomials help streamline the factorization process, making it quick and effective.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. This is akin to breaking down numbers into prime factors but applied to polynomial expressions. Factoring is crucial for simplifying expressions and solving equations, especially when dealing with quadratic polynomials.

There are different strategies for polynomial factorization:

• Finding common factors: Look for terms that can be evenly divided by a common value.
• Factoring by grouping: Used for polynomials with four or more terms.
• Factoring completely: Breaking a polynomial down into its irreducible factors.
• Recognizing special products: Such as the difference of squares and perfect square trinomials.

Mastering polynomial factorization is an essential skill in algebra that enables solving and simplifying algebraic expressions efficiently.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation signs. They serve as the fundamental building blocks in algebra, representing quantities in abstract form.

Understanding and manipulating algebraic expressions are key skills in math, allowing for the solving of equations and inequalities. This includes:

• Combining like terms: Simplifying expressions by adding or subtracting terms with the same variable.
• Applying the distributive property: Rewriting expressions by multiplying a single term across a sum or difference within parentheses.
• Simplifying complex expressions: Reducing expressions to their simplest form.

Algebraic expressions are used to model real-life situations in various disciplines, making them a versatile and powerful tool in the mathematician's toolkit. They form the basis for more complex equations and functions used in advanced mathematics and applied sciences.