Factoring Trinomials with a ≠ 1 04:54 minutes

Video Transcript

Transcript Factoring Trinomials with a ≠ 1

Today is a big day for Samuel. He wants to join the baseball team, but since he's not a real big athlete he uses his engineering skills to figure out a way to make the team.
Samuel developed a special pair of glasses to analyze his opponents. He uses factoring trinomials to figure out when the ball will hit the ground. Before the player hits the ball, he gets a function of the flight path of the ball to know where it will land. Let's take a look at the view from his glasses. Samuel is standing in right field, ready for anything.

Analyzing the Function

The batter hits the ball with this function: h(x) = -5x² + 14x + 3, where h(x) represents the height of the ball in feet and 'x' represents the time in seconds. We are looking for h(x) = 0, or when the ball hits the ground. Hmm this is not easy to solve, but Sammy's special spectacles can factor things in a jiffy.
As you know, the standard form of a quadratic function is y = ax² + bx + c. In this problem, a = -5, b = 14, and c = 3. First, we have to find the product of a and c, which is -5 times 3, which equals -15. Now, we have to find the factor pairs of -15 (since -15 is negative, only one of the factors should be negative).
Let's think of the possible factors of -15. The factors we have are -1(15), 1(-15), -3(5), and 3(-5). These are the only possible factor pairs of -15. Next, we want to find the pair that sums to b, which, in our case is positive 14. Let's look at the sum of these factors. Remember, we're looking for 14 as the result. The first combination of -1 and 15 equals 14. It looks like we've found the correct factor pair.

Factorization of the trinomial term

To factor this quadratic function, where the right side is a trinomial with 'a ≠ 1', we use the box method. We fill the box with the terms of the quadratic function. Generally, we place the first term in the upper-left-hand corner and the last term in the lower-right-hand corner. Now that we know that 14x is equal to -1x + 15x, we can complete the box with these terms in the other two corners.
We need to find the greatest common factor for each row. For the top row, the greatest common factor is -1x. For the bottom row, the greatest common factor is 3. Next we need to find the greatest common factor for each column. For the first column, the greatest common factor is 5x and 1 for the second. The factorization of the trinomial term is the product of the GCF column sum (5x + 1) and the GCF row sum (-1x + 3) giving us (5x + 1)(-x + 3). We can write our trinomial as a product of two binomials.

Setting h(x)=0

Because we want to know where the ball hits the ground, we set h(x) = 0 because 'h' represents the height of the ball. The equations 5x + 1 = 0 and -x + 3 = 0 represent the two times the ball will hit the ground. The first step is to subtract the integer value from both sides of the equation. The next step is to divide by the coefficient in front of 'x' on both sides of the equation. The solutions to the two equations are -1/5 and 3. We can't use -1/5 because we can't measure time with negative values, so the ball will hit the ground after 3 seconds.

With the help of his glasses, Samuel is in position at the correct time. But maybe he should practice catching a little more...