Question

Factor each polynomial. $$ x^{2}+7 x+10 $$

Short answer

(x + 2)(x + 5)

Step-by-step solution

Identify the polynomial

Recognize that the polynomial to be factored is \(x^2 + 7x + 10\). This is a quadratic polynomial of the form \(ax^2 + bx + c\), with \(a = 1\), \(b = 7\), and \(c = 10\).

Find two numbers that multiply to \(c\) and add to \(b\)

Look for two numbers that multiply to \(10\) (the constant term) and add up to \(7\) (the coefficient of the linear term). The numbers \(2\) and \(5\) both multiply to \(10\) and add up to \(7\).

Write the polynomial as a product of binomials

Using the numbers from step 2, express the polynomial as a product of two binomials: \((x + 2)(x + 5)\).

Verify the factored form

Expand \((x + 2)(x + 5)\) to confirm it equals the original polynomial: \((x + 2)(x + 5) = x^2 + 5x + 2x + 10 = x^2 + 7x + 10\). Since they match, the factored form is correct.

Key concepts

Quadratic Polynomials

Let's start by understanding what a quadratic polynomial is. Quadratic polynomials are expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents a variable. The defining feature of a quadratic polynomial is the \(x^2\) term, which dictates that the highest power of the variable is 2.
An example is \(x^2 + 7x + 10\). Here, \(a = 1\), \(b = 7\), and \(c = 10\). Quadratics often come up in various fields like physics, engineering, and finance to model parabolic paths and trends.
To solve quadratic polynomials, we perform factoring, complete the square, or use the quadratic formula. Factoring is often the simplest of these when it’s applicable.

Factoring Quadratics

Factoring quadratics is an important skill that can make solving these equations much easier. The concept involves breaking down the quadratic into simpler polynomials that multiply together.
For instance, take our example \(x^2 + 7x + 10\). In this case, you want to find two numbers that multiply to the constant term \(c\) (which is 10) and add up to the coefficient of the linear term \(b\) (which is 7). The numbers 2 and 5 fit these criteria.
Therefore, you can write the original quadratic as a product of two binomials: \((x + 2)(x + 5)\). This allows you to break down more complex expressions into more manageable parts.

Binomials

A binomial is a polynomial with exactly two terms. For example, \((x + 2)\) and \((x + 5)\) are binomials. When we factor a quadratic polynomial like \(x^2 + 7x + 10\), we express it as a product of two binomials.
This gives us an easier way to solve equations and perform further calculations.
In our specific example, we end up with \((x + 2)(x + 5)\). Each of these binomials can be visualized as a building block of the original quadratic expression.
This approach lets us work with simpler components, streamlining the solving process.

Polynomial Multiplication

Polynomial multiplication is a fundamental concept that helps verify if our factoring is correct. To multiply polynomials, you use the distributive property.
In our example, we factor out \(x^2 + 7x + 10\) as \((x + 2)(x + 5)\). To confirm it's accurate, multiply the binomials together:

• First, multiply \(x\) with \(x\), giving \(x^2\).
• Then, multiply \(x\) with 5, giving \(5x\).
• Next, multiply 2 with \(x\), giving \(2x\).
• Lastly, multiply 2 with 5, yielding 10.

Add these together: \(x^2 + 5x + 2x + 10\), which simplifies to \(x^2 + 7x + 10\), matching our original polynomial.
Polynomial multiplication not only checks our work but also deepens our understanding of how binomials combine to form quadratics.