Problem 14 - Elementary and Intermediate Algebra

Question

CONCEPTS Use the substitution \(x=a+b\) to rewrite the trinomial \(6(a+b)^{2}-17(a+b)-3\).

Step-by-Step Solution

Answer

The trinomial is rewritten as \(6x^2 - 17x - 3\).

Step 1: Perform Substitution

Let \(x = a + b\). Substitute \(x\) into the expression: \(6(a+b)^{2}-17(a+b)-3\) becomes \(6x^{2} - 17x - 3\).

Step 2: Expand the Expression

Expand \(6x^{2} - 17x - 3\). Notice the expression is already expanded.

Step 3: Simplify the Expression

Since the expression is already in simplified form \(6x^{2} - 17x - 3\), there is no further simplification needed.

Key Concepts

Trinomials

A trinomial is a type of polynomial expression that consists of three terms. In algebra, trinomials often appear in the form of quadratic expressions, which include terms with squared variables alongside linear and constant terms. Typically, you will see trinomials written as:

• ax² + bx + c

Where:

• a, b, and c are coefficients, with a ≠ 0
• ax² is the quadratic term
• bx is the linear term
• c is the constant term

Understanding trinomials is crucial because they form the basis for solving many algebraic equations and are commonly seen throughout various math problems, including applications in real-world scenarios. Trinomials can be factored, solved using the quadratic formula, or rewritten using algebraic substitution, which is a technique to simplify expressions by introducing new variables.

Polynomial Expressions

Polynomial expressions are mathematical phrases that consist of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. In a more generalized definition, polynomials can have one or more terms expressed as:

• Each term is made up of a coefficient (a number) multiplied by the variable(s) raised to a whole number exponent.
• They are classified based on their number of terms: monomials (one term), binomials (two terms), trinomials (three terms), and so on.
• The highest exponent is called the degree of the polynomial.

Polynomial expressions, such as trinomials, play a prominent role in algebraic manipulations. To simplify complex polynomials, algebraic substitution is often used, which is a technique of replacing certain expressions within the polynomial with simpler variables. This simplifies the process of solving and expanding expressions, making polynomial properties easier to analyze and understand.

Quadratic Equations

Quadratic equations are important in algebra and are defined as second-degree equations of the form:

• ax² + bx + c = 0

Here, 'a,' 'b,' and 'c' are constants, with 'a' not equal to zero. Quadratic equations can have different solutions depending on the values of these coefficients. They can either:

• Have two distinct real solutions
• Have one real solution (repeated root)
• Have two complex solutions

To solve quadratic equations, several methods can be employed:

• Factoring, which involves writing the equation as a product of binomials
• Using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Completing the square, a method of transforming the equation into a perfect square trinomial
• Graphing to visually identify the roots of the equation

Understanding quadratic equations is essential for dealing with polynomial expressions as they frequently occur in algebra and calculus problems.