Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{2}+11 x+10 $$

Short answer

(x + 1)(x + 10)

Step-by-step solution

Identify the Polynomial

Given the polynomial: \[ x^2 + 11x + 10 \]

Find Two Numbers that Multiply to the Constant Term

Look for two numbers that multiply to 10 (the constant term) and add up to 11 (the coefficient of the linear term, x).

Identify the Numbers

The numbers 1 and 10 multiply to 10 and add up to 11.

Write out the Factored Form

Rewriting the polynomial as a product of binomials using the numbers identified: \[ x^2 + 11x + 10 = (x + 1)(x + 10) \]

Check the Factored Form

Expand \((x + 1)(x + 10)\) to ensure it equals the original polynomial: \[ (x + 1)(x + 10) = x^2 + 10x + x + 10 = x^2 + 11x + 10 \]

Key concepts

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler components, known as factors. These factors, when multiplied together, yield the original polynomial. It's similar to finding prime factors of a number.
For example, in the polynomial \(x^2 + 11x + 10\), we aim to express it as a product of simpler polynomials.
This process helps solve equations and simplify expressions, making complex problems easier to handle.
To factor a polynomial, you need to identify patterns or apply specific techniques like grouping, using the quadratic formula, or factoring by inspection.

Binomials

A binomial is a polynomial with exactly two terms. It's often written as \(a + b\) or \(a - b\).
In the example \(x^2 + 11x + 10\), we factor it into two binomials: (x + 1) and (x + 10).
When factoring, we look for numbers that multiply to give the constant term (here, 10) and add up to the coefficient of the linear term (here, 11).
This approach simplifies many algebraic expressions and equations, making it an essential skill in algebra.
Remember, the key is identifying pairs of numbers that meet these criteria, then writing the polynomial as the product of these binomials.

Quadratic Equations

A quadratic equation is a second-degree polynomial, typically written as \(ax^2 + bx + c = 0\).
The given polynomial, \(x^2 + 11x + 10\), is a quadratic equation with \(a = 1\), \(b = 11\), and \(c = 10\).
Factoring quadratics often helps solve these equations by setting each factor equal to zero and solving for x.
For our example, once factored as (x + 1)(x + 10), we find solutions by setting each binomial to zero:
\(x + 1 = 0\) -> \(x = -1\)
\(x + 10 = 0\) -> \(x = -10\).
Therefore, the solutions to the equation \(x^2 + 11x + 10 = 0\) are x = -1 and x = -10.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols.
In algebra, we use letters (like x and y) to represent numbers in equations and expressions.
The process of solving and simplifying these equations often requires operations such as factoring, expanding, and simplifying polynomials.
In our example, we used algebraic techniques to factor the polynomial \(x^2 + 11x + 10\) into simpler terms.
Mastery of these techniques forms the foundation for more advanced topics in mathematics, engineering, physics, and many other fields.
Understanding how to manipulate and solve polynomial equations improves problem-solving skills and logical thinking.