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

Mathematics

2 answers:

S_A_V [24]3 years ago

8 0

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.