Problem 48 - Elementary Algebra

Question

For the following problems, factor the polynomials, if possible. $$ s^{2}+9 s+18 $$

Step-by-Step Solution

Answer

Question: Factor the given polynomial: $$s^2 + 9s + 18$$ Answer: The factored form of the given polynomial is $$(s+3)(s+6)$$.

Step 1: Identify the coefficients

Identify the coefficients from the given quadratic polynomial, which is in the form $$ax^2 + bx + c$$. Here, a = 1, b = 9, and c = 18.

Step 2: Find the factors

We will find two factors of 18 that, when added, equal 9. The factors of 18 are (1, 18), (2, 9), and (3, 6). Among these, the factors that add up to 9 are 3 and 6.

Step 3: Write the polynomial as the product of two binomials

Now that we have found the correct factors, we can write the polynomial as a product of two binomials: $$(s+3)(s+6)$$

Step 4: Final Answer

The given quadratic polynomial, after factoring, is: $$(s+3)(s+6)$$

Key Concepts

Quadratic Polynomial

A quadratic polynomial is a second-degree polynomial most commonly written in the standard form \( ax^2 + bx + c \), where \( a \), \( b \) and \( c \) are constants, with \( a \) not equal to zero. It represents a parabolic graph on a coordinate plane, featuring a characteristic 'U' shape, and can intersect the x-axis at zero, one, or two points, which are also known as the roots.Identifying the structure of a quadratic polynomial is essential in the process of factoring. In our exercise \( s^2 + 9s + 18 \), the coefficients are right in front of us: \( a = 1, b = 9 \) and \( c = 18 \). Recognizing these values provides a base from which to begin the process of factoring the quadratic expression, turning what may seem a daunting expression into a solvable puzzle.

Product of Binomials

The product of binomials follows the pattern often referred to as FOIL (First, Outer, Inner, Last) which stands for combining the first terms, the outer terms, the inner terms, and the last terms of each binomial. When factoring a quadratic polynomial, we are essentially doing this process in reverse to find two binomials whose product gives the original quadratic polynomial.For instance, the given problem requires us to rewrite the quadratic polynomial \( s^2 + 9s + 18 \) as the product of binomials. After determining suitable factors of the constant term (in this case, 18) that when added together give us the middle term (the coefficient of \( s \) which is 9), we arrive at the binomials that solve our problem: \( (s+3)(s+6) \) which correspond to the 'First' and 'Last' terms of the FOIL method. Our choice of 3 and 6 for factoring \( 18 \) ensures that when we expand the product of these two binomials, we'll recover our original polynomial.

Factors of a Number

Factors of a number are integers that can be multiplied together to produce that number. For any positive integer \( n \), the set of factors always includes at least 1 and \( n \), with possibly several pairs in between. To solve our factoring exercise, focusing on the constant term \( c \), which is 18 in this case, we explore all pairs of factors that multiply to give us 18: \( (1, 18), (2, 9), (3, 6) \).Identifying pairs of factors that add up to the coefficient of the middle term (the \( b \) value, which is 9 for our exercise) is a crucial step. After listing the factors, we see the pair (3, 6) not only multiplies to give us 18 but also adds up to give us 9, thus satisfying both conditions needed for successful factoring of the quadratic polynomial into two binomials. This critical observation helps to create a clear connection between the factors of a number and the process of factoring quadratic polynomials.