Use synthetic division to perform each division. (3x^3+6x^2-8x+3)/(x+3)

Question

Accepted Answer

Hey, everyone in this problem, we're asked to perform synthetic division. The expression we're given is five X cubed plus eight X squared minus 11 X plus 370 all divided by X plus five. We're given four answer choices. Option A five X squared minus 17 X plus 74. Option B five X squared minus 17 X minus 74. Option C five, X squared minus 17 X plus plus 370 divided by X plus five. And option D five, X squared minus 17 X plus minus 370 divided by X plus five. Now, we're gonna start by drawing our synthetic division table and we're gonna start by filling out the term on the far left. Now, on the far left, we have to write what we're dividing by. OK? And in our synthetic division table that comes from the root of the dominator. OK. So we have our denominator XX plus five, we're gonna set this to equal to zero And we get that X is equal to negative five. OK? This is the root of the denominator. And so we use that value of -5 on the left of our table. Now we're gonna fill in the rest of the top rope and that's gonna come from the coefficients of our numerator. OK. On the left of our table, we're gonna start with the coefficient corresponding to the highest exponent term. We're gonna work our way all the way to the constant term on the far, right. So our highest coe or sorry, our highest exponent is three. The coefficient corresponding to that term is five and we have five execute. Next, we have our X squared term The coefficient of eight, then we have our X term with a coefficient of negative And finally, our constant term of 370. Now, in order to perform the synthetic division, we're gonna start on the left-hand side and we're gonna bring that five down through our table and we're gonna write it underneath. When we have a value underneath in our table, we multiply it up To this value on the left hand side. five multiplied by negative five Gives us -25. We're gonna write that in the next column over and just above our table from above the bottom of our table. Sorry. Now we have a complete column. So we're gonna add within the column eight plus negative 25 gives us negative 17. And we write that down below. Now we have a new value below our table. When we have a value below our table, we multiply it with the value on the left hand side of our table, -17, multiplied by negative five gives us positive 85. We write it in the next column over and just above the bottom of our table. Now we have a complete column again. Now we're gonna add within that column. -11-plus 85 gives us 74. We write it down below And again, we have a new value below our table. We multiply it to the value on the left of our table 74 multiplied by negative five gives us negative 370. And then we have a complete column and we add 370-plus negative 370 gives us the zero. Right now. We need to interpret the results. So this final number on the far right hand side underneath our table is our remainder. In this case, the remainder is zero. So when we do that division, we get a polynomial, no remainder. Now, what does the polynomial look like? Wow, We started with a polynomial of degree three And we divide it by a polynomial of degree one. And when we divide poli by a polynomial of degree one, it reduces the degree of the polynomial by one. OK. So we started with three, we're reducing it by one. So we're gonna get a polynomial of degree two. OK? A quadratic. So starting on the left hand side of a resulting table. And underneath our table, these correspond to the coefficients of our resulting polynomial on the left. It corresponds to the highest exponent term just like it did when we filled out the first row. So This is a degree two polynomial. The highest exponent is gonna be two. And so five is the coefficient that corresponds with the X squared term. OK. Our next coefficient is gonna be negative 17. That's gonna correspond with our X term. So we have five X squared -17x. And then finally, the next value working our way to the right 74 corresponds with our constant term. So we add 74. OK. Our remainder zero. So that's it. And the result of this division is five X squared minus 17 X plus 74 which corresponds with answer choice A, that's it for this one. Thanks everyone for watching. I hope this video helped.

Use synthetic division to perform each division. (3x^3+6x^2-8x+3)/(x+3)

Verified Solution

Key Concepts

Synthetic Division

Polynomial Coefficients

Remainder Theorem