Question about roots/synthetic division

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In summary, according to the synthetic division shown, the original polynomial is 5x^3+ 6x^2+ 7x+ 6 and the number -1 is a root of this polynomial. This is because when using synthetic division, the actual root or zero of the polynomial is put out to the left, and the polynomial function contains the factor $(x-r)$. This may be why there was confusion about the sign of the root. Synthetic division is a quick way of dividing a polynomial by x- a, and the remainder is P(a) when x= a. In this case, the synthetic division shows that when x= -1, the value of the polynomial is 0, indicating that -1 is
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According to the synthetic division done below, what was the original polynomial and what number do we know is a root of that polynomial? Explain how you know to receive full points.

I know what the polynomial is, but I thought the root of this polynomial would be 1 but it's actually -1 could somebody explain why?
 

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When doing synthetic division, the actual root $r$ or zero of the polynomial is put out to the left. Thus we know:

\(\displaystyle f(r)=0\)

And the polynomial function will contain the factor $(x-r)$. This might be why you felt the number needed to be negated?
 
  • #3
Your question is puzzling. Are you clear on what "synthetic division" is? It is a quick way of dividing a polynomial by x- a for some value of a. Specifically, we write just the coefficients of the polynomial (here that is "5 6 7 6" which tells us that the original polynomial was \(\displaystyle 5x^3+ 6x^2+ 7x+ 6\). The number we are "dividing" by is a= -1. In general, if we divide a polynomial, P(x), by x- a, the remainder is P(a).

The synthetic division here, shows that when x, in this polynomial, is set to -1, the value of the polynomial. is 0. That means that x= -1 is a root.
 

1. What is synthetic division and how is it different from long division?

Synthetic division is a method used to divide polynomials. It is similar to long division in that it involves dividing one polynomial by another, but it is much faster and more efficient than long division. Instead of writing out the entire division process, synthetic division only requires writing out a few numbers and performing simple arithmetic operations.

2. When is synthetic division typically used?

Synthetic division is typically used when dividing by a linear polynomial, meaning a polynomial of the form ax + b. It is also commonly used when finding the roots, or solutions, of a polynomial equation.

3. Can synthetic division be used for both finding roots and dividing polynomials?

Yes, synthetic division can be used for both purposes. When using it to divide polynomials, the results of synthetic division will give the quotient and remainder. When using it to find roots, the last number in the resulting row will be the root of the polynomial equation.

4. What are the steps for performing synthetic division?

The steps for performing synthetic division are as follows:

  1. Write the coefficients of the polynomial being divided.
  2. Write the root of the divisor (the number being divided by) to the left of the coefficients.
  3. Bring down the first coefficient.
  4. Multiply the root by the first coefficient, and write the result under the next coefficient.
  5. Add the two numbers in the same column and write the result below them.
  6. Repeat steps 4 and 5 until all coefficients have been used.
  7. The resulting numbers will be the coefficients of the quotient (except for the last number, which will be the remainder).

5. Are there any limitations to using synthetic division?

Yes, there are limitations to using synthetic division. It can only be used when dividing by a linear polynomial, and the divisor must have a leading coefficient of 1. If the divisor does not have a leading coefficient of 1, it must be factored or divided out before using synthetic division.

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