Consider the function 𝑓 of 𝑥 equals 𝑥 to the fourth power plus five 𝑥 cubed minus eight 𝑥 squared plus nine. What does the remainder theorem tell us about 𝑓 of three?
The remainder theorem tells us about the remainder when we divide. So what would it tell us about 𝑓 of three? According to the remainder theorem, 𝑓 of three will be the remainder when we divide our function by 𝑥 minus three.
And we’re told that, “Hence use synthetic division to find 𝑓 of three.”
So we need to use synthetic division to find this remainder. So finding 𝑓 of three means we’re taking our function and dividing it by 𝑥 minus three. But really if we set 𝑥 minus three equal to zero, we would need to add three to both sides of the equation. So we get three. So we will be actually dividing by three which is the same as 𝑥 minus three. So three goes on the outside.
And now, for the function, it’s the coefficients and the constants that go on the inside. But we can’t miss anything. So in front of 𝑥 to the fourth, we have a one. And in front of 𝑥 cubed, we have a positive five. And in front of 𝑥 squared, we have a negative eight. However, notice that there is no 𝑥. So we need to add in a zero. And then, next would be the constant, which is nine. So it’s really important not to forget that zero if there’s a term missing.
So in order to begin, we bring the one down and then we take one times three which is three. And we add the column together. So five plus three is eight and we start over. Eight times three is 24 and negative eight plus 24 is 16. And now, we take 16 times three and we get 48. Zero plus 48 is 48. And now, 48 times three is 144 and nine plus 144 would be 153.
So the number all the way to the right is the remainder. And then, the next number will be the constant. And then, the next number would be the coefficient of 𝑥. And eight would be the coefficient of 𝑥 squared. And one would be the coefficient of 𝑥 cubed.
So 𝑓 of three we said would be the remainder. Therefore, 𝑓 of three, using the remainder theorem, is equal to 153.