Question Video: Finding the Zeros of a Function by Factoring Mathematics • Third Year of Preparatory School

Video Transcript

Find, by factoring, the zeros of the function 𝑓 of 𝑦 is equal to 𝑦 squared plus eight 𝑦 plus seven.

In this question, we’re given a function 𝑓 of 𝑦, which we can see is a quadratic function in 𝑦. We’re asked to find the zeros of this function, and we’re told we need to do this by using the method of factoring. To answer this question, let’s start by recalling what we mean by the zeros of a function. The zeros of any function 𝑓 of 𝑦 are the values of 𝑦 such that 𝑓 evaluated at 𝑦 is zero. In other words, they’re the input values of the function which output zero.

There are many different ways of determining the zeros of a function. We’re asked to do this by using the method of factoring. To use this method of factoring, let’s start by setting up our equation. We need to set our function 𝑓 equal to zero. This gives us the equation 𝑦 squared plus eight 𝑦 plus seven is equal to zero. We need to solve this equation for 𝑦. To factor a quadratic equation where the coefficient of 𝑦 squared is one, we can do this by finding a pair of numbers which multiply to give us the constant term, that’s seven, and add to give us the coefficient of 𝑦, that’s eight.

To find such a pair of numbers, the easiest way is to start by listing all of the numbers we know which multiply to give us the constant term. These are called the factor pairs of the number. In this case, there’s only one such pair. One times seven is equal to seven. And we can notice the sum of these two numbers, one plus seven, is equal to the coefficient of 𝑦. That’s eight. Factoring then tells us we can factor the quadratic as follows. Since seven times one is equal to seven and seven plus one is equal to eight, 𝑦 squared plus eight 𝑦 plus seven is equal to 𝑦 plus seven multiplied by 𝑦 plus one.

And we could verify this if we wanted to by distributing the parentheses. We can do this by using the FOIL method. Multiplying the first two terms, we get 𝑦 times 𝑦, which is 𝑦 squared. Next, multiplying the outer two terms, we get one multiplied by 𝑦, which is 𝑦. Then, the product of the inner terms is seven 𝑦. Seven 𝑦 plus 𝑦 is equal to eight 𝑦. Finally, the product of the last two terms is equal to seven. This means we’ve successfully factored our quadratic.

Now, we can see on the left-hand side of our equation, we have a product and we can see that this product must be equal to zero. And for a product to be equal to zero, one of the factors must be equal to zero. Therefore, either 𝑦 plus seven is equal to zero or 𝑦 plus one is equal to zero. We can solve both of these equations for 𝑦. First, to solve 𝑦 plus seven is equal to zero, we subtract seven from both sides of the equation to get 𝑦 is equal to negative seven. Next, to solve 𝑦 plus one is equal to zero, we subtract one from both sides of the equation. We get 𝑦 is equal to negative one. Therefore, by factoring, we were able to find the zeros of the function 𝑓 of 𝑦 is equal to 𝑦 squared plus eight 𝑦 plus seven. The zeros were negative seven and negative one.