Problem 21 - Elementary Algebra

Question

Factor the following problems, if possible. $$ 10 x^{2}+33 x+20 $$

Step-by-Step Solution

Answer

Answer: The factors of the quadratic expression 10x^2 + 33x + 20 are (5x + 4) and (2x + 5).

Step 1: Identify the quadratic expression

First, let's identify the given quadratic expression that needs to be factored: $$ 10x^2 + 33x + 20 $$

Step 2: Find the product of the leading coefficient and the constant term

Next, find the product of the leading coefficient (10) and the constant term (20): $$ 10 \times 20 = 200 $$

Step 3: Find two numbers that multiply to 200 and add up to 33

Now, we need to find two numbers that multiply to 200 and add up to 33, which are the coefficients of the x terms when factored. In this case, the values are 25 and 8 because: $$ 25 \times 8 =200 $$ and, $$ 25 + 8 = 33 $$

Step 4: Rewrite the quadratic expression using the new coefficients of x

Now, let's rewrite the quadratic expression using the new coefficients of x (25 and 8): $$ 10x^2 + 25x + 8x + 20 $$

Step 5: Factor by grouping

Now, let's factor the quadratic expression by grouping the terms: $$ 5x(2x + 5) + 4(2x +5) $$

Step 6: Extract the common factor

Lastly, extract the common factor (2x + 5) from both terms: $$ (5x + 4)(2x + 5) $$ So, the factored form of the given quadratic expression is: $$ (5x + 4)(2x + 5) $$

Key Concepts

Leading Coefficient

When looking at a quadratic expression, the leading coefficient is an essential aspect to consider. It is the number in front of the highest degree term, which is generally the square term in a quadratic. In the expression \(10x^2 + 33x + 20\), the leading coefficient is \(10\). This number can influence the methods you use to factor the quadratic expression.

• The larger the leading coefficient, the more challenging factoring can become.
• It's crucial as its multiplication with other terms might require you to find specific pairs of numbers.

Recognizing the leading coefficient will help you understand how other terms relate to each other when rewriting expressions for factoring.

Factoring by Grouping

Factoring by grouping is a technique used to simplify the process of breaking down polynomials into simpler, factorable expressions. After setting up a new four-term polynomial, you can group pairs of terms separately for factoring. For instance, from the expression \(10x^2 + 25x + 8x + 20\), we group it as \( (10x^2 + 25x) + (8x + 20) \).

• Identify pairs of terms that can be factored together.
• Extract common factors from each group to create a new expression.

In this case, we find common factors within each pair, resulting in \(5x(2x + 5) + 4(2x + 5)\). This shows how sometimes simplifying into shared factors can easily lead to finding the overall factored form by extracting common binomials.

Quadratic Expression

A quadratic expression is essentially a mathematical phrase involving terms up to a squared degree. Typically, it takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Take, for instance, \(10x^2 + 33x + 20\).

• The term \(10x^2\) represents the quadratic term because of the square of \(x\).
• The term \(33x\) is the linear term, and follows the square term in degree.
• The number \(20\) is the constant term in this context.

Understanding this structure is vital because it sets the foundation for how a quadratic can be factored or solved using various algebraic methods.

Constant Term

The constant term in a quadratic expression is the term without any variables, serving as the fixed part of the expression. In our example \(10x^2 + 33x + 20\), the constant term is \(20\).

• This term is important for product considerations during the factoring process.
• It helps in determining the constant factors when factoring by grouping or other methods.

In the factoring process, we find the product of the leading coefficient and the constant term (\(10 \times 20 = 200\)). This helps in determining how the middle term (\(33x\)) can be splitted using factors of this product. Hence, grasping the role of the constant term is crucial in making successful, strategic moves in factoring quadratics.