Question

Please answer this polynomials question. Thank You!

Answer 1

Thank you for your question.

Question:-

When $x^{19}+2x^{14}+3x^{9}+4x^{4}+5$ is divided by $x^{5}-x^{4}+x^{3}-x^{2}+x-1$, the remainder is $a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$. Find the value of $a_{0}+a_{1}+a_{2}+a_{3}+a_{4}$.

Method:-

The key is that the value is the remainder at $x=1$.

Also, the division algorithm holds true for all values of $x$. We know that polynomial division works in the following process.

$P(x)=D(x)Q(x)+R(x)$

• $P(x)$ means the first letter of 'polynomial.'
• $D(x)$ means the first letter of 'divisor.'
• $Q(x)$ means the first letter of 'quotient'
• $R(x)$ means the first letter of 'remainder.'

The previous equation is always satisfied for all values of $x$. Such equations are called the identities.

Solution:-

If we factorize the divisor, we see $x-1$ as a factor.

$x^{5}-x^{4}+x^{3}-x^{2}+x-1=(x^{2}+x+1)(x^{2}-x+1)(x-1)$

This means if we substitute $x=1$ to the given equation, the $D(x)$ becomes zero.

So,

$P(1)=D(1)Q(1)+R(1)$ and $D(1)=0$.

$\implies P(1)=R(1)$.

So where $x=1$,

$R(1)=(1)^{19}+2(1)^{14}+3(1)^{9}+4(1)^{4}+5=15$.

The value of $a_{0}+a_{1}+a_{2}+a_{3}+a_{4}=\boxed{15}$.

Solve advanced problems:-

Question:-

The remainder when $f(x)$ is divided by $(x-1)^{2}$ is $2x-1$, and when divided by $x-1$ is $1$. Find the remainder when divided by $(x-1)^{2}(x-3)$.

Answer: $x^{2}-4x+2$

Answer key:-

• The remainder is a quadratic polynomial.
• The remainder when divided by $(x-1)^{2}$ is found by dividing $ax^{2}+bx+c$, since $(x-1)^{2}(x-3)Q(x)$ is divisible by $(x-1)^{2}$.

Solution:-

Since $(x-1)^{2}(x-3)$ is a cubic polynomial, let the remainder for division by

The following equation is an identity.

$\implies f(x)=(x-1)^{2}(x-3)Q(x)+ax^{2}+bx+c$

If we divide $ax^{2}+bx+c$ by $(x-1)^{2}$, since we are given that the remainder is $2x-1$, the following is an identity. Since given that the remainder is $2x-1$,

$\implies ax^{2}+bx+c=a(x-1)^{2}+2x-1$.

Since given that $f(3)=1$,

$\implies f(3)=(2)^{2}a+2(3)-1$

$\implies f(3)=4a+5$

$\implies 4a+5=1$

$\implies a=-1$.

Hence the required answer is $-(x-1)^{2}+2x-1$, or $\boxed{-x^{2}+4x-2}$.