Problem 66 - Elementary Algebra

Question

For the following problems, factor the polynomials, if possible. $$ 3 y^{4}-27 y^{3}+24 y^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial is \(3y^2(y - 1)(y - 8)\).

Step 1: Identifying common factors in the terms of the given polynomial

We are given the polynomial: $$ 3y^4 - 27y^3 + 24y^2 $$ We can see that all the terms have a common factor of 3 and 'y'. Let's first factor out the common term.

Step 2: Factor out the common term

Factoring out the common term 3 and 'y' from the polynomial, we get: $$ 3y^2(y^2 - 9y + 8) $$

Step 3: Observe the resulting quadratic expression

After factoring out the common term, we are left with the quadratic expression \((y^2 - 9y + 8)\). Now we must factor this quadratic expression.

Step 4: Factoring the quadratic expression

In order to factor the quadratic expression, we should look for two numbers that multiply to 8 and add to -9. We find that -1 and -8 are such numbers. Therefore, we can rewrite the quadratic expression in the form \((y - 1)(y - 8)\). This gives us: $$ 3y^2(y - 1)(y - 8) $$ This is the factored form of the given polynomial.

Key Concepts

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of variables raised to an exponent and multiplied by coefficients. These expressions can vary in complexity depending on the number of terms they have and the degree of the exponents. Commonly, polynomials are classified based on their number of terms and degree:

• A monomial is a polynomial with just one term, such as \(5x^3\).
• A binomial has two terms, for example, \(x^2 - 4\).
• A trinomial consists of three terms, like \(3y^4 - 27y^3 + 24y^2\).

Polynomials play a crucial role in mathematics, modelling real-world phenomena and serving as the building blocks for more complex expressions. In the given exercise, the expression \(3y^4 - 27y^3 + 24y^2\) is a trinomial since it has three terms. Understanding how to factor these expressions is essential for simplifying equations and solving problems effectively.

Quadratic Expressions

Quadratic expressions are a special type of polynomial expression where the variable is raised to the second power (the degree is 2). A standard form of a quadratic expression looks like \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Factoring quadratic expressions is a crucial skill as it simplifies the expression, making it easier to solve for the variables involved.

In our example, once we factored out the common factors, we were left with the quadratic portion \(y^2 - 9y + 8\). To successfully factor this quadratic expression, we follow a simple process that involves finding two numbers that multiply to the constant term 8 and add up to the linear coefficient, -9.

This process of trial and error helps us rewrite the quadratic expression as a product of two binomials, \((y - 1)\) and \((y - 8)\), which is much simpler to work with. Recognizing these patterns and practicing factoring will enhance your problem-solving skills.

Common Factors

Common factors are elements that are shared between all terms of an expression, and factoring them out is a fundamental technique in simplifying polynomials. Identifying common factors involves:

• Looking at the coefficients, which are the numerical parts of each term.
• Finding any variables that appear in every single term.

In the expression \(3y^4 - 27y^3 + 24y^2\), each term has a common factor of 3 and at least two 'y' variables. Factoring means dividing each term by these common factors and writing them outside of a set of parentheses. Consequently, the expression is transformed into \(3y^2(y^2 - 9y + 8)\), which makes further operations, like solving or simplifying, much easier.

Selecting the correct common factors is a skill that will improve with practice. By thoroughly understanding how to factor out the largest common factors, you can simplify and solve polynomial expressions more efficiently.