Problem 43 - Elementary Algebra

Question

Factor. $$ 20 a_{3} b_{2}-60 a_{2} b_{3}+45 a b_{4} $$

Step-by-Step Solution

Answer

The expression factors to \( 5 a b^2 (2a - 3b)^2 \).

Step 1: Identify the Greatest Common Factor (GCF)

To begin factoring the expression \( 20 a^3 b^2 - 60 a^2 b^3 + 45 a b^4 \), first identify the greatest common factor of the coefficients (20, 60, and 45). The GCF of these numbers is 5. Next, identify the common variables and their lowest powers. Each term has at least one \( a \) and two \( b \)'s, so the GCF for the variables is \( a b^2 \). Thus, the overall GCF is \( 5 a b^2 \).

Step 2: Factor out the GCF

Divide each term in the expression by the GCF \( 5 a b^2 \).- The first term: \( \frac{20 a^3 b^2}{5 a b^2} = 4 a^2 \)- The second term: \( \frac{-60 a^2 b^3}{5 a b^2} = -12 a b \)- The third term: \( \frac{45 a b^4}{5 a b^2} = 9 b^2 \)So the expression becomes: \( 5 a b^2 (4 a^2 - 12 a b + 9 b^2) \).

Step 3: Recognize the Quadratic as a Perfect Square

Observe the quadratic expression \( 4 a^2 - 12 a b + 9 b^2 \). Check if it can be expressed as a perfect square:\( 4 a^2 = (2a)^2 \) and \( 9 b^2 = (3b)^2 \). The middle term \( -12 a b = 2(2a)(3b) \) ensures that it is indeed a perfect square.Thus, the quadratic can be expressed as \( (2a - 3b)^2 \).

Step 4: Write the Final Factored Form

Combine the factored parts from previous steps to write the entire expression in factored form:\( 5 a b^2 (2a - 3b)^2 \). This is the completely factored expression of the original polynomial.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept in factoring polynomials. Finding the GCF helps simplify an expression by allowing you to factor out the largest number and variable powers common to each term. In the exercise provided, the polynomial is \( 20 a^3 b^2 - 60 a^2 b^3 + 45 a b^4 \).

First, identify the greatest number that can divide each coefficient (20, 60, and 45) evenly. This number is their GCF, and in this case, it is 5.
Also examine the variables. Each term in the polynomial includes at least one \( a \) and two \( b \)'s. Thus, for the variables, the GCF is \( a b^2 \).

• 20 can be expressed as \( 5 \times 4 \)
• 60 as \( 5 \times 12 \)
• 45 as \( 5 \times 9 \)

By factoring out the GCF \( 5 a b^2 \), the polynomial simplifies, making it easier to handle. It's crucial to always check for the GCF when beginning to factor a polynomial expression.

Perfect Square

Recognizing a perfect square trinomial is another important aspect of factoring polynomials. A perfect square trinomial is an algebraic expression that can be written as the square of a binomial. This means it takes the form \( (x+y)^2 \) or \( (x-y)^2 \).

In the step involving the quadratic part of the expression \( 4a^2 - 12ab + 9b^2 \), we check if it is a perfect square.
To do this, we see:

• \( 4a^2 \) is \( (2a)^2 \)
• \( 9b^2 \) is \( (3b)^2 \)
• The middle term \( -12ab \) is \( 2 \times (2a) \times (3b) \)

This checks out as a valid perfect square because the middle term fits the pattern \( 2xy \).
Thus, \( 4a^2 - 12ab + 9b^2 \) can be rewritten as \( (2a - 3b)^2 \). Understanding perfect squares helps in recognizing and simplifying complex algebraic forms.

Algebraic Expressions

Polynomials like the equation above are a type of algebraic expression. Working with algebraic expressions involves a set of skills crucial for understanding mathematics.

Let's review what makes up these expressions:

• **Terms:** Individual components, separated by plus or minus signs, involve variables like \( a \) and \( b \).
• **Coefficients:** Numbers multiplying the variable parts. In \( 20a^3b^2 \), 20 is the coefficient.
• **Variables and Exponents:** Variables (like \( a \) and \( b \)) often possess exponents, indicating how many times a variable is used as a factor.

Recognizing how these parts connect in expressions allows for methods such as factoring.
Algebraic expressions can be simplified or factored using the GCF as introduced earlier. Mastery of these components will enable successful manipulation and simplification of polynomials and other more complex expressions, making further mathematics much more approachable.