Question

If p(x) = 2x^3 - 3x + 5, what is the remainder of p(x) divided by (x - 5)

Answer 1

The remainder of $P\left( x \right) = 2{x^3} - 3x + 5$ divided by $\left( {x - 5}\right)$ is

Explanation:

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

The polynomial is $P\left( x \right) = 2{x^3} - 3x + 5$ and $\left( {x - 5} \right).$

The numerator of the division is $P\left( x \right) = 2{x^3} - 3x + 5$ and the denominator is

Solve the given polynomial $P\left( x \right) = 2{x^3} - 3x + 5$ by the use of synthetic division.

Now obtain the value of $x$ from the denominator.

$\begin{aligned}x - 5 &= 0\\x&= 5\\\end{aligned}$

Divide the coefficients of the polynomial by $5.$

$\begin{aligned}5\left| \!{\nderline {\,{2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\, - 3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5} \,}} \right. \hfill\\\,\,\,\,\underline{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,10\,\,\,\,\,\,\,\,\,\,\,\,\,50\,\,\,\,\,\,\,\,\,\,\,\,\,\,235}\hfill\\\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,10\,\,\,\,\,\,\,\,\,\,\,\,\,47\,\,\,\,\,\,\,\,\,\,\,\,\,\,240\hfill\\\end{aligned}$

The last entry of the synthetic division tells us about remainder and the last entry of the synthetic division is $240$.Therefore, the remainder of the synthetic division is $240.$

The remainder of $p\left( x \right) = 2{x^3} - 3x + 5$ divided by $\left( {x - 5}\right)$ is $\boxed{240}.$

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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

The question is

if p(x)= 2x³-3x+5, what is the remainder of p(x) divided by (x-5)

We use the synthetic division to find the remainder,

x-5=0

x=5

Now, write the coefficients of polynomial.

5 | 2  0  -3    5

   | ↓ 10  50  235

-------------------------------

     2 10  47  | 240→ is the remainder.

Quotient= 2x²+10x+47

Remainder= 240