Use synthetic division to perform each division. (x^4 + 4x^3 + 2x^2 + 9x+4) / x+4

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Hey, everyone in this problem, we're asked to perform synthetic division. We're given X to the exponent four plus six X cubed plus six, X squared plus 40 X plus divided by X plus six. We're given four answer choices. Option A X cubed plus six, X plus four minus five divided by X plus six. Option B X cubed plus nine X squared plus six, X plus four plus five divided by X plus six. Option C X cubed plus six, X plus four and option D X cubed plus nine X squared plus six, X plus four. Now we're gonna start by rewriting what we were given. So we have X to the exponent four plus six X cubed plus six, X squared plus 40 X plus 24 all divided by X plus six. Now, we're asked to do synthetic division. So let's set up our synthetic division table. Now, this table is gonna have two rows within it. We're gonna have a value on the left hand side and we're gonna start by filling out that value and that value on the left hand side is going to be the root of the denominator So the denominator of our division is X plus six. We're gonna set this equal to zero to find the root. And we get that X is equal to negative six and that value of negative six is going to go on the left hand side of our table. Now, we're gonna fill in the top row of our table. The top row is gonna come from the coefficients of the numerator. So the numerator is a polynomial we're gonna take, starting from the left, the coefficient corresponding to the highest exponent term and work our way all the way to the right with the constant term. So the highest exponent term in the numerator is X to the exponent four. It has a coefficient of one. Then we move to our next future that has a coefficient of six. Our X squared term has a coefficient of six. Our X term has a coefficient of 40. And finally, our constant term is 24. And so that top row of our table from left to right is 166, 40 24. Now we're gonna perform the division, we're gonna start with the value on the left hand side. This one, we bring it down and underneath our table. When we have a value underneath our table, we multiply it to the value on the left of the table. One multiplied by negative six gives us negative six. We write it in the next column over in the second row and then we're gonna add within that coal. So we have six plus negative six. This gives us zero. We write it down below. We have a value underneath our table. We multiply it to the value on the left zero, multiplied by negative six gives us zero. We write it in the next column over in the second row. And then we add in that coal six plus zero gives us six. We write it down below. We have a new value below. We multiply it to the value on the left six multiplied by negative six gives us negative 36. We write it in the next column over in the second row, we add up that column 40 plus negative gives us four. We write it down below. We multiply it to the value on the left four multiplied by negative six gives us negative 24. We write it in the next column over and we add up that column. This is our final column. We have 24 plus negative 24 which gives us zero. Now I'm gonna draw a dash line separating this last column. OK? And this last number underneath our table actually represents the remainder of our division. OK? So in this case, the remainder is zero. So when we do this division, we have no remainder. What that means is that we're gonna get a polynomial out of this. Now let's give ourselves some more room to write out what that polynomial is gonna be. Now, let's think about the degree of this polynomial. First, we're taking a polynomial of degree four and divided by a polynomial of degree one. When we divide by a polynomial of degree one, we reduce the degree of our polynomial by one. So we have X to the exponent four. Now we're gonna have an X Q that's gonna be our highest exponent term. OK. A polynomial of degree three. Now the values in the bottom row of this table are gonna represent the coefficients. And again, it's the same pattern starting on the left, we have the coefficient corresponding to the highest exponent term. And we work down to the right with the coefficient of the constant term. OK? So X cubed is our highest exponent term. It's gonna have a coefficient of one. So we get one X cubed, we move to the next term. It's gonna be our X squared term. It has a coefficient of zero plus zero X squared. Then we add our next term, we have a coefficient of six. Our next highest exponent is X or just one. Sorry, the exponent is one, the term is X. So we get six X and then finally, we have our constant term and that's just equal to four. So we get one X cubed plus zero X squared plus six X plus four. And we can write this as X cubed plus six X bless And so that is the result. When we perform the division using synthetic division, we found that the correct answer is X cubed plus six X plus four, which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

Use synthetic division to perform each division. (x^4 + 4x^3 + 2x^2 + 9x+4) / x+4

Verified Solution

Key Concepts

Synthetic Division

Polynomial Functions

Remainder Theorem