Question

If division of a polynomial by a binomial results in a remainder of zero, what can be concluded

Answer 1

If division of a polynomial by a binomial result in a remainder of zero we can conclude that the binomial is a factor of the polynomial.

Further Explanation:

If division of a polynomial by a binomial result in a remainder of zero means that the binomial is a factor of polynomial.

Consider a polynomial as $P\left( x \right) = {x^2} + 2x + 1$ and consider a binomial as $x +1.$

The numerator of the division is ${x^2} + 2x + 1$ and the denominator is $x +2.$

Solve the given polynomial $P\left( x \right) = {x^2} + 2x + 1$ by the use of synthetic division.

Now obtain the value of x from the denominator.

$\begin{aligned}x + 1&= 0\\x&=- 1\\\end{aligned}$

Divide the coefficients of the polynomial by- 1, to check whether - 1 is a zero of the polynomial.

$\begin{aligned}- 1\left| \!{\nderline {\,{1\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1} \,}} \right. \hfill \\\,\,\,\,\,\,\underline {\,\,\,\,\,\,\,\,\,\,\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\, - 1\,} \hfill \\\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \hfill \\\end{aligned}$

The last entry of the synthetic division tells us about remainder and the last entry of the synthetic division is 0. Therefore, the remainder of the synthetic division is 0.

Therefore,$x+2$ is a factor of ${x^2} + 2x + 1.$

If division of a polynomial by a binomial result in a remainder of zero we can conclude that the binomial is a factor of the polynomial.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Synthetic Division

Keywords: division, binomial synthetic division, long division method, coefficients, quotients, remainders, numerator, denominator, polynomial, zeros, degree.

Answer 2

Answer:

Binomial is a factor of the polynomial.

Step-by-step explanation:

You can divide a polynomial with a binomial by using synthetic division in the form x - c. We can use this to find several things: the quotient and the remainder you get when you divide the polynomial function by x - c.

For a binomial which divides a polynomial completely, giving a remainder of zero and some quotient then it means that the binomial is a factor of that polynomial.