Question

Factor each polynomial. $$ x^{2}+9 x+8 $$

Short answer

\((x + 1)(x + 8)\)

Step-by-step solution

Identify the Coefficients

For the quadratic polynomial in the form of \(ax^2 + bx + c\), identify the coefficients:\(a = 1\), \(b = 9\), and \(c = 8\).

Find Two Numbers that Multiply to c and Add to b

Search for two numbers that multiply to \(c = 8\) and add up to \(b = 9\). The numbers are 1 and 8 because \(1 \times 8 = 8\) and \(1 + 8 = 9\).

Write the Polynomial as a Product of Two Binomials

Re-write the polynomial in terms of the factors identified: \(x^2 + 9x + 8 = (x + 1)(x + 8)\).

Double-Check the Factors

Expand \((x + 1)(x + 8)\) to ensure it matches the original polynomial. \((x + 1)(x + 8) = x^2 + 8x + 1x + 8 = x^2 + 9x + 8\). This confirms the factoring is correct.

Key concepts

quadratic equations

Quadratic equations are a type of polynomial equation that come in the form \(ax^2 + bx + c = 0\). The highest exponent is a square (hence 'quadratic'), making the graph of these equations a parabola. The coefficients \(a\), \(b\), and \(c\) are usually real numbers.To understand how to solve these equations, you might use methods like 'factoring', the 'quadratic formula', or 'completing the square'.
For example, consider the quadratic equation from the exercise: \(x^2 + 9x + 8\). By identifying \(a = 1\), \(b = 9\), and \(c = 8\), we can use factoring to solve it. Finding the correct pairs of factors plays a crucial role here because we are finding values that could provide roots of the equation.

polynomial factorization

Polynomial factorization is the process of breaking down a polynomial into simpler terms (factors) that when multiplied, recreate the original polynomial. It's like breaking down a number into its prime factors but for polynomial expressions.
For the quadratic polynomial \(x^2 + 9x + 8\), we start by finding two numbers that multiply to \(c = 8\) and add up to \(b = 9\). These numbers are 1 and 8. So, we re-write the polynomial as \((x + 1)(x + 8)\). This decomposition simplifies solving or analyzing the equation.
Always double-check your factors by expanding them back out to ensure you return to the original polynomial. In our case, expanding \((x + 1)(x + 8)\) brings us back to \(x^2 + 9x + 8\), confirming our factorization is spot on.

binomials

A binomial is a type of polynomial that contains exactly two terms. These terms are generally combined using the plus (+) or minus (-) signs. It's a simpler polynomial and often used in factoring larger polynomials.
Take the exercise polynomial \(x^2 + 9x + 8\). When factored, it breaks down into two binomials: \((x + 1)\) and \((x + 8)\). Each of these binomials contributes to building the original quadratic polynomial when multiplied together.
Understanding binomials helps in simplifying polynomials and equations. It’s also foundation knowledge for more advanced algebra concepts. So next time you encounter a polynomial, look for opportunities to break it down into binomials!