Problem 9 - Elementary Algebra

Question

Factor, if possible, the following trinomials. $$ m^{2}-8 m+16 $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial is $$(m - 4)^2$$.

Step 1: Inspect the trinomial

We are given the trinomial $$m^2 - 8m + 16$$, and we want to factor it, if possible, into two binomials \((m-a)(m-b)\).

Step 2: Determine the factors of 16

First, we need to find the factors of 16 to determine the potential values for a and b. The factors of 16 are: $$1, 2, 4, 8, 16$$ and their corresponding negative values.

Step 3: Find values of a and b that sum to -8

We need to find a pair of factors from step 2 that will add up to the middle term of the trinomial, which is -8. From our list of factors, we can see that $$-4 + (-4) = -8$$, so $$a = b = -4$$.

Step 4: Write the factored trinomial

Knowing the values for a and b, we can now write the trinomial in its factored form: $$(m - 4)(m - 4)$$.

Step 5: Simplify the factored trinomial

Since the two binomials are the same, we can rewrite the factored trinomial as: $$(m - 4)^2$$. The factored form of the given trinomial is $$(m - 4)^2$$.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). These equations are called "quadratic" because the highest power of the variable \(x\) is squared. Understanding the properties of quadratic equations is crucial for solving various problems in algebra. To solve these equations, one can use several methods, such as:

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

Quadratic equations can have two solutions, which can be real or complex numbers depending on the discriminant \(b^2 - 4ac\). In our exercise, the quadratic trinomial \(m^2 - 8m + 16\) simplifies into a perfect square \((m - 4)^2\), indicating that both solutions are real and equal.

Polynomials

Polynomials are algebraic expressions made up of variables and coefficients bonded together through operations of addition, subtraction, multiplication, and non-negative integer exponents. They take a general form of \(a_nx^n + a_{n-1}x^{n-1} + \,\ldots\, + a_1x + a_0\), where \(a_n, a_{n-1},\ldots,a_1, a_0\) are constants and \(n\) is a non-negative integer. The polynomial used in our work is a quadratic trinomial, meaning it has three terms: \(m^2 - 8m + 16\). Each term's degree is associated with the power of the variable "\(m\)." Quadratics are specifically polynomials where the highest degree term is two.Understanding polynomials:

• Degree: It indicates the highest exponent with a non-zero coefficient, in this case, 2 for \(m^2\).
• Classification by terms: Quadratics like the one in our problem have three terms, known as trinomials.

Trinomials like \(m^2 - 8m + 16\) often arise in algebra and are frequently factored to reveal simpler expressions or find roots of equations.

Factored Form

The factored form of a polynomial involves expressing it as a product of two or more simpler polynomials. Factoring offers valuable insights into the roots of the polynomial and aids in solving equations more efficiently. For a quadratic polynomial like \(m^2 - 8m + 16\), the goal is to express it in the form \((m-a)(m-b)\), simplifying its analysis.In this specific case, we identify \(m^2 - 8m + 16\) as a perfect square trinomial, leading to \((m-4)^2\). This factored form points out that \(m = 4\) is the repeated root. Understanding factored forms involves:

• Recognizing patterns, such as perfect squares and differences of squares, which make factoring easier.
• Using rules of algebra to simplify expressions.

Breaking down polynomials into their factored form simplifies problem-solving. Once factored, they allow for straightforward equation solutions and can simplify wider algebraic expressions.