Question

Factor each polynomial. $$ x^{2}+7 x+6 $$

Short answer

(x + 1)(x + 6)

Step-by-step solution

- Identify the Coefficients

Identify the coefficients of the quadratic polynomial. For the given polynomial, the coefficients are: - Coefficient of \(x^2\) is \(1\).- Coefficient of \(x\) is \(7\).- Constant term is \(6\).

- Find the Factors of the Constant Term

List all pairs of factors of the constant term (6) that add up to the middle coefficient (7). The pairs are: - (1, 6) - (2, 3) Since \(1 + 6 = 7\), we identify (1, 6) as the correct pair.

- Write the Factors

Use the identified pair to write the polynomial as a product of two binomials. \(x^2 + 7x + 6 = (x + 1)(x + 6)\)

- Verify the Factors

Expand the product \((x + 1)(x + 6)\) to verify it equals the original polynomial. \((x + 1)(x + 6) = x^2 + 6x + x + 6 = x^2 + 7x + 6\).The factorization is correct.

Key concepts

quadratic polynomial

A quadratic polynomial is an expression in the form of \[ax^2 + bx + c\] where:

• \(a\)

  is the coefficient of \(x^2\),

• \(b\)

  is the coefficient of \(x\),

• \(c\)

  is the constant term.

Quadratic polynomials have the highest degree of 2 among their terms. This means that the highest exponent of the variable (here, \(x\)) is 2.
Quadratic polynomials form the basis for much of algebra. Understanding how to factor them is essential for solving quadratic equations, graphing parabolas, and in many real-world applications such as physics and engineering.

coefficients

Coefficients are the numerical factors in the terms of a polynomial. For the quadratic polynomial \[x^2 + 7x + 6\], the coefficients are:

• \(1\text{ (for } x^2\text{)}\),
• \(7\text{ (for } x\text{)}\),
• \(6\text{ (the constant term)}\).

Coefficients are crucial because they determine the shape and direction of a parabola when the quadratic polynomial is graphed.
When factoring or solving quadratic polynomials, we focus on manipulating these coefficients to find simpler expressions that are easier to work with. This can also help in understanding how different terms interact within a polynomial.

binomials

Binomials are algebraic expressions that contain exactly two terms. Factoring quadratic polynomials often involves breaking them down into a product of two binomials. For instance, the polynomial \(x^2 + 7x + 6\) can be factored into the binomials\[(x + 1)(x + 6)\].Each binomial is simpler to work with compared to the original quadratic polynomial.
To factor the quadratic polynomial effectively:

• Identify pairs of numbers whose product equals the constant term,
• and whose sum equals the coefficient of the \(x\text{ term}.\).

These pairs guide the formation of binomials, streamlining calculations in solving equations or other algebraic operations.

constant term

In a polynomial, the constant term is the number without any variable attached to it. For the quadratic polynomial \(x^2 + 7x + 6\), the constant term is \(6\).
The constant term plays a significant role in factoring quadratics, as you need to find two numbers:

• whose product equals the constant term,
• and whose sum equals the coefficient of the middle term (in this case, 7).

Understanding how to manipulate the constant term helps simplify the factorization process, making it easier to break down complex polynomials into simpler factors.