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Question: How do you use synthetic division to divide \(\dfrac{{10{x^4} - 50{x^3} - 800}}{{x - 6}}\)?...

How do you use synthetic division to divide 10x4−50x3−800x−6\dfrac{{10{x^4} - 50{x^3} - 800}}{{x - 6}}?

Explanation

Solution

In this problem, we have to divide the given polynomial by using another polynomial. It means a division of a polynomial. In this problem, we are going to use the synthetic division of polynomials. It is the simplest method to divide the polynomial by using another polynomial.

Complete step-by-step solution:
In this problem, we are going to divide the given polynomial by another polynomial
We are going to use synthetic to divide the polynomials.
Synthetic division is the simplest method to divide the polynomials.
In this problem, the given polynomials are
Let we take
p(x)=10x4−50x3−800p\left( x \right) = 10{x^4} - 50{x^3} - 800and
q(x)=x−6q\left( x \right) = x - 6
Now we have to divide the polynomial p(x)p\left( x \right)by the polynomial q(x)q\left( x \right) by using the synthetic division.
Synthetic division is used when the divisor in the form of x±a[(x−a)or(x+a)]x \pm a\left[ {\left( {x - a} \right)or\left( {x + a} \right)} \right]
That’s like that we have the divisor, q(x)=x−6q\left( x \right) = x - 6
In this method of synthetic division, we only use the coefficients of the dividend.
Therefore we get
We get, x=6x = 6
And for the divide bring the leading coefficients down.
Here, arrange the coefficients from the leading term, we arrange from the leading term at the polynomial p(x)p\left( x \right),
Therefore the coefficient of x2,x{x^2},x is zero.
x4=10 x3=−50 x2=0 x=0{x^4} = 10 \\\ {x^3} = - 50 \\\ {x^2} = 0 \\\ x = 0
And the constant term is −800 - 800
Here we notice that the given polynomial does not have the terms x2,x{x^2},x
Therefore
6∣! 10−5000−800   ‾6{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| \\!{\underline {\, 10{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 50{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 800 \\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\\ \,}} \right.
At the next step write the leading term as same and multiple the leading coefficient by the divider 66
Now, we get
6∣! 10−5000−800 0  ‾ 106{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| \\!{\underline {\, 10{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 50{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 800 \\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \\\ \,}} \right. \\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 10
Now multiply the leading term one by the divider66, and put the value in the send column, under the next leading term.
6∣! 10−5000−800 060603602160  ‾ 10106036013606{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| \\!{\underline {\,10{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 50{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - 800 \\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 60{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 60{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 360{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2160 \\\ \,}} \right. \\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 10{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 10{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 60{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 360{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 1360
Now we solved the problem, by using the synthetic division method.
The last term 13601360 is the remainder of the division.
And the quotient is,
The first term 1010 is the coefficient of x3{x^3}
The second term 1010 is the coefficient of x2{x^2}
The third one 6060 is the coefficient of xx
And the constant term is 13601360
Hence the quotient is 10x3+10x2+60x+36010{x^3} + 10{x^2} + 60x + 360
Hence the solution for this problem is 10x3+10x2+60x+360,136010{x^3} + 10{x^2} + 60x + 360,1360

Note: Synthetic division is a shortcut that can be used to divide the polynomial when the divisor is in the form x±ax \pm a. And only the coefficients of the given polynomial are used for the synthetic division. Use the bottom numbers to write the quotient of the polynomial. The number which is in the last column is called the remainder. And the next number has degree zero and the next number has degree one and the next number has degree two and so on.