Find Zeros of $2x^3+5x^2-28x-15$ with Synthetic Division

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  • Thread starter karush
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In summary, the conversation discussed the use of synthetic division and the quadratic formula to factor a polynomial $g(x)$. The result was $g(x) = (x+10)(x+\frac{1}{2})(x-3)$ with roots at $-10, -\frac{1}{2}, 3$. However, there may be an error in the application of the quadratic formula, as the coefficient for $x^2$ should be 2 instead of 1.
  • #1
karush
Gold Member
MHB
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$\text{factor}$
$g(x)=2x^3+5x^2-28x-15$
$\text{synthetic division}$
$\begin{array}{c|rr}
& 2 & 5 &-28 &-15 \\
&&6&33 & 15\\ \hline
3&2&11&5&0
\end{array}$
$\text{thus }\\$
$2x^2+11x+5$
$\text{use quadradic formula}$
$\begin{align*}
x=\frac{-(11)\pm\sqrt{(11)^2-4(2)(5)}}{2(1)}
&=\frac{-11\pm\sqrt{81}}{2}
=\frac{-11\pm9}{2}
\end{align*}$
$x=-10,-1 \quad g(x)=(x+10)(x+1)(x-3)$
hopefully
comment?
 
Last edited:
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  • #2
Check your application of the quadratic formula, or simply factor. :)
 
  • #3
a=2 not 1

use quadradic formula
$\begin{align*}
x=\frac{-(11)\pm\sqrt{(11)^2-4(2)(5)}}{2(2)}
&=\frac{-11\pm\sqrt{81}}{4}
=\frac{-11\pm9}{4}
\end{align*}$
$\text{then $x=-5,-\frac{1}{2}$
so
$g(x)=(x+10)(x+\frac{1}{2})(x-3)$}$
 
Last edited:

Related to Find Zeros of $2x^3+5x^2-28x-15$ with Synthetic Division

1. What is synthetic division?

Synthetic division is a method used to divide a polynomial by a linear expression. It is a shortcut method that allows for the determination of the zeros of a polynomial without having to use long division.

2. How do you perform synthetic division?

To perform synthetic division, you will need to set up the division in a table, with the coefficients of the polynomial in one row and the root of the polynomial in the second row. Then, follow the steps of synthetic division, which include dividing the first coefficient by the root, multiplying that result by the root, and then subtracting that product from the next coefficient. The final result will be the remainder, and the coefficients in the last row will be the coefficients of the quotient.

3. Why use synthetic division to find zeros?

Synthetic division is helpful in finding the zeros of a polynomial because it is a quicker and more efficient method than long division. It also allows for the easy identification of the remainder, which can provide information about the polynomial, such as whether a given number is a root or not.

4. How does synthetic division help to find zeros?

Synthetic division helps to find zeros by determining the values for which the polynomial is equal to zero. If the remainder is equal to zero, then the number used as the root is a zero of the polynomial. The coefficients in the final row can also be used to write out the simplified form of the polynomial, making it easier to identify the zeros.

5. Can synthetic division be used to find zeros for all polynomials?

No, synthetic division can only be used to find zeros for polynomials with real coefficients and a degree of three or higher. It also only works for linear factors, meaning the root must be a linear expression. For polynomials with complex or irrational roots, other methods such as the quadratic formula or graphing may need to be used to find the zeros.

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