Synthetic Division P(x)|2+3i= 0

[shorty888]

P(x)= x^4-4x^3+10x^2+12x-39, using synthetic division given 2+3i is a zero of function

 

[CaptainBlack]

P(x)= x^4-4x^3+10x^2+12x-39, using synthetic division given 2+3i is a zero of function

Exactly what do you want to do, what you have posted is not a question. Please post the question as asked.

CB

 

[Opalg]

P(x)= x^4-4x^3+10x^2+12x-39, using synthetic division given 2+3i is a zero of function

If a polynomial with real coefficients has a complex zero, then the complex conjugate of that number is also a zero. Thus 2+3i and 2-3i are both zeros. By the factor theorem, $x-(2+3i)$ and $x-(2-3i)$ are both factors of $P(x)$. Hence so is their product $\bigl(x-(2+3i)\bigr)\bigl(x-(2-3i)\bigr)$. Work out that product (which is a real quadratic polynomial), then use synthetic division to divide $P(x)$ by that quadratic. The quotient will be another quadratic, which you can solve to get the other two zeros of $P(x)$.

 

[HallsofIvy]

My understanding of synthetic division is that it is used to divide by "x- a" for a constant a, not a quadratic. Of course, it is true that
$$(x-(2-3i))(x+(2- 3i))= ((x- 2)- 3i)((x-2)+ 3i)= (x-2)^2- (3i)^2= x^2- 4x+ 4+ 9= x^2- 4x+ 13$$ divides into $$x^4- 4x^3+ 10x^2+ 12x- 39$$ without remainder but synthetic division by x- (2+ 3i) is2+3i|1_____-4_______10_________12_______-39
__________2+3i_____-13_______-6+9i_______+39
____1____-2+3i______-3________6+9i________0

or $$x^3+ (-2+3i)x^2- 3x+ (6+ 9i)$$

 

[CellCoree]

Synthetic division is a useful tool in solving polynomial equations, especially when dealing with complex numbers. In this case, we are given that 2+3i is a zero of the function P(x)= x^4-4x^3+10x^2+12x-39. This means that when we plug in 2+3i into the function, the result will be 0.

To use synthetic division, we first set up the division problem by writing the coefficients of the polynomial in descending order. In this case, the polynomial is already in descending order, so we can proceed with the division. The divisor will be (x-2-3i), which is the factor form of 2+3i.

We then bring down the first coefficient, which is 1. Next, we multiply this coefficient by the divisor, (x-2-3i), and write the result under the next coefficient. In this case, we get 1(x-2-3i)=x-2-3i. We then add this result to the next coefficient, -4, and write the sum under the next coefficient. This process is repeated until we reach the last coefficient.

After completing the division, we get a remainder of 0, which confirms that 2+3i is indeed a zero of the function. The quotient obtained from the division, in this case, is x^3-2x^2+3x-13. This means that we can rewrite the original function as P(x)=(x-2-3i)(x^3-2x^2+3x-13).

Using this information, we can find the other zeros of the function by setting the quotient equal to 0 and solving for x. This will give us the remaining three zeros of the function, which are all real numbers.

In conclusion, synthetic division is a powerful tool in solving polynomial equations, especially when dealing with complex numbers. By using this method, we can easily find the zeros of a given function and rewrite it in factored form, making it easier to solve and understand.