Problem 63 - Elementary and Intermediate Algebra

Question

Factor out \(-1\) from each polynomial. $$ 9-4 a $$

Step-by-Step Solution

Answer

The factored expression is \(-1(-9 + 4a)\).

Step 1: Identify the Expression

The expression given here is \(9 - 4a\). Our task is to factor out \(-1\) from this polynomial.

Step 2: Factor \(-1\) from the Polynomial

To factor \(-1\) from the polynomial, multiply the entire expression by \(-1\) but change the signs of each term within the expression. This means that each term within the parenthesis should have the opposite sign. So, \(9\) becomes \(-9\) and \(-4a\) becomes \(+4a\).

Step 3: Write the Final Expression

The expression after factoring out \(-1\) is: \[-1 \times (-9 + 4a)\].This can be simplified to \(-1(-9 + 4a)\).

Key Concepts

Understanding Polynomials

Polynomials are versatile mathematical expressions that are composed of variables and coefficients, linked together using arithmetic operations like addition, subtraction, multiplication, and non-negative integer exponents. A simple polynomial can look like this: \(x^2 + 3x + 2\).

To help you grasp polynomials better, here are a few key points to remember:

• They're classified based on the number of terms they have: a monomial with one term, a binomial with two terms, and a trinomial with three terms.
• The degree of a polynomial is the highest exponent of its variable.
• Polynomials are a cornerstone in algebra, giving us a foundation for further exploration in mathematics.

As you manage operations with polynomials, you'll find that they often need to be manipulated for simplification or solving purposes.

Negative Factorization Explained

Negative factorization can sound a bit intimidating at first, but it's simply the process of taking out a negative sign from a polynomial. This means you'd factor out \(-1\) from each term, effectively switching all the signs in the expression.

Here's how to visualize it:

• Identify each term in the polynomial. For example, in \(9 - 4a\), you have \(9\) and \(-4a\).
• Multiply each term by \(-1\), changing positive numbers to negative, and negative numbers to positive.
• For the polynomial \(9 - 4a\), factoring out \(-1\) results in \(-9 + 4a\).

Factoring out negative numbers can simplify problem-solving in algebra, making it easier to see solutions and relationships between terms.

Decoding Algebraic Expressions

Algebraic expressions contain numbers, variables, and operators that signify mathematical operations. Think of them as the language of algebra, expressing relationships and calculations that can be solved.

In tackling algebraic expressions, keep these insights in mind:

• They don't have an equals sign, unlike equations. This distinguishes expressions from algebraic equations.
• Simplifying expressions often involves combining like terms or using distributive properties.
• Factoring is a key technique in both simplifying expressions and solving equations.

By mastering the manipulation of algebraic expressions, like factoring out \(-1\), you'll improve your problem-solving skills, setting a strong foundation for more advanced algebra and calculus.