Question Video: Finding the Set of Zeros of a Polynomial Function Mathematics • 10th Grade

Find the set of zeros of the function ๐‘“(๐‘ฅ) = 7๐‘ฅโถ โˆ’ 112๐‘ฅโด.

02:22

Video Transcript

Find the set of zeros of the function ๐‘“ of ๐‘ฅ equals seven ๐‘ฅ to the sixth power minus 112๐‘ฅ to the fourth power.

Weโ€™re asked to find the set of zeros of this function ๐‘“ of ๐‘ฅ which is a polynomial. The zeros or roots of a polynomial ๐‘“ of ๐‘ฅ are the values that ๐‘ฅ equals ๐‘Ž such that ๐‘“ of ๐‘Ž equals zero. Essentially, they are the values of the variable ๐‘ฅ that make the value of the function zero.

To find the zeros of ๐‘“ of ๐‘ฅ then, we set the function equal to zero. We then observe that both terms have a shared factor of ๐‘ฅ to the fourth power. So we can factor by ๐‘ฅ to the fourth power to give ๐‘ฅ to the fourth power multiplied by seven ๐‘ฅ squared minus 112 equals zero. Furthermore, as seven is a factor of 112, we can also factor by seven to give seven ๐‘ฅ to the fourth power multiplied by ๐‘ฅ squared minus 16 equals zero. We now have that the product of two factors is equal to zero.

It follows that at least one of the individual factors must be equal to zero. And so we have seven ๐‘ฅ to the fourth power equals zero, or ๐‘ฅ squared minus 16 equals zero. We can now solve these equations separately to find the zeros of ๐‘“ of ๐‘ฅ.

The first equation can be simplified by dividing both sides by seven. And then, if ๐‘ฅ to the fourth power is equal to zero, it follows that ๐‘ฅ itself must be equal to zero. To solve the second equation, we begin by adding 16 to both sides to give ๐‘ฅ squared equals 16. ๐‘ฅ is then equal to plus or minus the square root of 16, which is positive or negative four.

Weโ€™ve found all solutions to these equations, and so weโ€™ve found all the zeros of ๐‘“ of ๐‘ฅ. Weโ€™re asked to give the answer as a set. So we have that the set of zeros of the function ๐‘“ of ๐‘ฅ is the set containing the values zero, four, and negative four.

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