Problem 23 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 5 . \(-9 x+20+x^{2}\)

Step-by-Step Solution

Answer

The trinomial factors to \((x - 5)(x - 4)\).

Step 1: Arrange the trinomial in descending order

The given trinomial is \(-9x + 20 + x^2\). First, we need to rewrite this in standard form (descending powers of \(x\)): \(x^2 - 9x + 20\).

Step 2: Identify coefficients

In the trinomial \(x^2 - 9x + 20\), identify the coefficients: \(a = 1\), \(b = -9\), \(c = 20\). This follows the format \(ax^2 + bx + c\).

Step 3: Determine two numbers that multiply to c and add to b

We need to find two numbers that multiply to \(c = 20\) and add to \(b = -9\). The numbers that meet these criteria are \(-5\) and \(-4\), since \((-5) \times (-4) = 20\) and \((-5) + (-4) = -9\).

Step 4: Express the middle term using the found numbers

Rewrite the middle term \(-9x\) using the two numbers found. The expression becomes \(x^2 - 5x - 4x + 20\).

Step 5: Factor by grouping

Group the terms in pairs: \((x^2 - 5x) + (-4x + 20)\). Factor out the common factors from each group. From the first group, factor out \(x\): \(x(x - 5)\). From the second group, factor out \(-4\): \(-4(x - 5)\).

Step 6: Factor the common binomial

Since both terms include the common binomial \((x - 5)\), factor it out: \((x - 5)(x - 4)\).

Step 7: Verify the factorization

Multiply \((x - 5)(x - 4)\) to check: \((x - 5)(x - 4) = x^2 - 4x - 5x + 20 = x^2 - 9x + 20\). This matches the original trinomial.

Key Concepts

Polynomial Expressions

Polynomial expressions are mathematical phrases that consist of variables and coefficients, connected by arithmetic operations of addition, subtraction, and multiplication. They can be as simple as a single number or variable, or more complex involving multiple terms like our trinomial example, \[-9x + 20 + x^2\]. Typically, these expressions show variables in different powers.

Here are important features of polynomial expressions:

• Each distinct term in a polynomial is composed of a coefficient (a constant number) and a variable raised to a whole number exponent.
• The degree of the polynomial is determined by the highest power of the variable present in the expression.
• Polynomial expressions are often written in standard form, which we'll discuss further.

When handling polynomial expressions, we often want them organized in descending powers of the variable, which makes operations like factoring and solving equations more manageable.

Factoring by Grouping

Factoring by grouping is an essential strategy used to simplify polynomials and make them easier to handle, often used when dealing with quadratics or other four-term polynomials. This method involves reorganizing and grouping terms to factor out common factors, eventually leading to the factorization of the entire expression.

To execute factoring by grouping, follow these fundamental steps:

• Arrange terms in a strategic way that allows grouping into pairs.
• Factor out the greatest common factor from each pair of terms.
• Identify and factor out the common binomial factor that emerges from the grouped terms.

In the exercise, when we applied factoring by grouping to \(x^2 - 9x + 20\), we restructured it and regrouped terms to simplify the expression to \((x-5)(x-4)\). This shows the power of the technique in breaking down complex problems into simpler factors.

Quadratic Equations

Quadratic equations are a type of polynomial equation that involves terms up to the second degree, typically written in the form \(ax^2 + bx + c = 0\). They are foundational in algebra and appear in various mathematical problems and real-world applications. Our previous trinomial example \(x^2 - 9x + 20\) is a quadratic expression.

Here are some key elements and characteristics of quadratic equations:

• The term \(ax^2\) is known as the quadratic term due to the variable being squared.
• The term \(bx\) is linear, involving the variable to the power of one.
• The constant term \(c\) is the value without a variable.

Solving quadratic equations can be done by methods like factoring, completing the square, or using the quadratic formula. Factoring, as applied in our example, especially benefits from identifying two numbers that multiply to the constant term and add up to the linear coefficient, aiding in breaking down the equation.

Standard Form of Polynomials

The standard form of a polynomial is an orderly way of writing polynomial expressions, where terms are arranged from highest to lowest power of the variable. For a quadratic polynomial, it would typically mirror the expression format \(ax^2 + bx + c\), as seen in the exercise after rearranging the terms of the trinomial.

Some features and steps to convert polynomials into standard form include:

• Identify all terms by their degree or highest power of the variable.
• Arrange them in decreasing order, starting with the term having the highest power of the variable.
• Re-order the polynomial if needed, ensuring consistency in notation and simplifying future calculations or factorizations.

Writing polynomials in standard form is crucial as it helps to easily identify the coefficients needed for solving or factoring, like in our example \(x^2 - 9x + 20\), making it easier to handle mathematically.