Given f(x) = 2x^3 - 7x^2 + 9x - 3, use the Remainder Theorem to f...

Question

Given f(x) = 2x^3 - 7x^2 + 9x - 3, use the Remainder Theorem to find f(- 13).

Accepted Answer

Hello. Today we're going to solve the polynomial expression using the remainder theorem. So the polynomial expression given to us is eight X. To the fourth power minus 17 X cubed minus 12 X squared plus nine X plus four. And we are asked to solve for F a five. Using the remainder theorem. Well the remainder theorem basically tells us that we want to go ahead and perform synthetic division along this polynomial expression using our input value, which in this case is five. So let's go ahead and set up our synthetic division first. Go ahead and draw some of the synthetic division symbol which is just this upside down division symbol. Next you're gonna want to put your input value five to the left hand side. And what goes inside the synthetic division symbol are the coefficients of all your terms of the polynomial. So the coefficients in this case are going to be 8 -17 -12, positive nine and positive four. And the very last term is going to be our remainder after performing the synthetic division which will also be our solution. So how do we perform the synthetic division? Well, the first thing you're gonna wanna do is bring down your first coefficient without altering it at all. So you're just gonna go ahead and bring down your first term which in this case is eight. Now what you wanna do is go ahead and multiply your first term with your input value. And that value is going to get added to the next term in the synthetic division. So eight times five is going to give us 40 And we're gonna add 42,. Doing so is going to give us the value of positive 23. Now you want to go ahead and repeat that pattern. We're gonna go ahead and multiply 23 with positive five and add that value to negative 1223 times five gives us the value of 115 And 115 -12 is going to give me 103. Now, once again, we're going to multiply 103 with positive five. That is going to give us the value of 515 And 515 9 is going to give me the value of 524. And finally we're gonna repeat this last step positive five times 524 and add that to positive four. So 524 times five gives us the value of 2620 and 2620 plus four is going to give us the value Of 2624. So what does this last number represent? Well, this is going to be the remainder over synthetic division. But since we're asked to solve for f of five, using the remainder theorem, that means that this number is going to be our solution. So our solution for f of five is going to be 2,624. which also means that the solution to this problem is going to be be so. I hope this video helps you in understanding how to do the remainder theorem, and I'll go ahead and see you all in the next video.

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