Which expression is the factorization of x2 + 10x + 21

Factorization

To factorize the quadratic expression $x^2 + 10x + 21$, we need to find two binomials that multiply together to give the original expression.

Finding Factors

We are looking for two numbers that multiply to $21$ (the constant term) and add up to $10$ (the coefficient of the linear term). The numbers that satisfy these conditions are $3$ and $7$, since:

Factoring the Expression

Using these numbers, we can break down the middle term $10x$ into $3x + 7x$. This allows us to write the expression as a sum of four terms:

Next, we group the terms into pairs and factor out the greatest common factor (GCF) from each pair:

We now see that there is a common binomial factor $(x + 3)$:

Final Factored Form

Therefore, the factorization of $x^2 + 10x + 21$ is:

Concept of Quadratic Expression Factorization

Factorizing a quadratic expression involves breaking it down into simpler expressions (products of binomials) that, when multiplied together, give the original quadratic expression. A general quadratic expression is of the form:

Here, $a$, $b$, and $c$ are constants. The goal of factorization is to express this quadratic expression as a product of two binomials.

Steps to Factorize

1. Identify coefficients: Note the coefficients $a$, $b$, and $c$ from the quadratic expression.

2. Find two numbers: These numbers should multiply to $ac$ (product of the coefficient of $x^2$ and the constant term) and add to $b$ (coefficient of $x$).

3. Rewrite the middle term: Use the two numbers found to split the middle term $bx$ into two terms.

4. Factor by grouping: Group the terms into pairs and factor out the greatest common factor (GCF) from each pair.

5. Combine terms: Express the quadratic expression as a product of two binomials.

Example

Consider the quadratic expression:

• Coefficient identification: $a = 6$, $b = 11$, and $c = 3$
• Find two numbers: We need numbers that multiply to $6 \times 3 = 18$ and add to $11$. The numbers $9$ and $2$ satisfy these conditions.
• Rewrite middle term: Split $11x$ as $9x + 2x$:

• Factor by grouping: Group the terms:

Factor out the GCF from each group:

• Combine terms: Factor out the common binomial:

Therefore, the factorized form of $6x^2 + 11x + 3$ is:

Important Points to Remember

• Check the product and sum of the pair of numbers carefully to ensure they fit the form $ac$ and $b$.
• Always factor out the GCF first if a common factor exists for all terms in the quadratic expression.

Special Cases

Perfect Square Trinomial:

If the quadratic expression is a perfect square trinomial, like $x^2 + 6x + 9$, it factors to:

Difference of Squares:

If the quadratic expression is a difference of squares, like $x^2 - 9$, it factors to:

Explanation

Finding common factors is a fundamental concept in mathematics often required in problems involving division, simplification of fractions, and number theory. A factor of a number is an integer that can divide that number without leaving a remainder. When finding common factors of two or more numbers, we identify those integers that can divide all the given numbers without leaving a remainder.

Steps to Find Common Factors

1. List the Factors of Each Number: Begin by listing all the factors of each given number.

   For example, to find the common factors of 12 and 18:

   • Factors of 12: 1, 2, 3, 4, 6, 12
   • Factors of 18: 1, 2, 3, 6, 9, 18

2. Identify Common Factors: Look for factors that appear in all lists.

   From our example:

   • Common factors of 12 and 18: 1, 2, 3, 6

Example with LaTeX Display

Let's find the common factors of 24 and 36.

1. List of Factors:

   • Factors of $24: 1, 2, 3, 4, 6, 8, 12, 24$
   • Factors of $36: 1, 2, 3, 4, 6, 9, 12, 18, 36$

2. Identify Common Factors:

Important Formulas

To express the greatest common factor (GCF), which is the largest of the common factors, between two numbers $a$ and $b$:

In our example, the GCF of 24 and 36 is $12$, as it is the largest number that divides both without leaving a remainder.

Practical Importance

Knowing common factors:

• Simplifies Fractions: The fraction $\frac{24}{36}$ can be simplified by dividing the numerator and the denominator by their GCF, 12, to get $\frac{2}{3}$.
• Solves Diophantine Equations: These are equations that require integer solutions, and finding common factors can simplify finding those solutions.
• Reduces Computational Complexity: In algorithms, especially those in cryptography, reducing the numbers to their common factors can make computations more manageable.

Conclusion

Understanding how to find common factors is essential for simplifying mathematical expressions and solving more complex problems efficiently.