The Remainder Theorem

When a certain number of things are divided into groups with an equal number of things in each group, the number of things left is known as the remainder. This is what “remains” after division.

Step by step guide to the remainder theorem

The remainder theorem is expressed as follows:

When a polynomial \(a(x)\) is divided by a linear polynomial \(b(x)\) whose zero is \(x = k\), the remainder is obtained by \(r = a (k)\). The remaining theorem enables us to compute the remainder of dividing any polynomial by a linear polynomial without performing the steps of the division algorithm.

Remainder theorem formula

The general formula of the remainder theorem is given as follows:

\(\color{blue}{p(x)=(x-c).q(x)+r(x)}\)

When \(p(x)\) is divided by \((x-a)\): remainder\(=p(a)\)

OR

When \(p(x)\) is divided by \((ax+b)\): remainder\(=p\left(-\frac{b}{a}\right)\)

Notes:

• When a polynomial \(a(x)\) is divided by a linear polynomial \(b(x\)) whose zero is \(x = k\), the remainder is given by \(r = a(k)\)
• The remainder theorem formula is: \(p(x)=(x-c).q(x)+r(x)\)
• The basic formula to check the division is: Dividend \(=\) (Divisor \(×\) Quotient) \(+\) Remainder

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The Remainder Theorem, Example 1:

Find the remainder when \(p(x)=3x^5-x^4+x^3-4x^2+2\) is divided by \(q(x):x-1\).

Replace the zero of \(q(x)\) into the polynomial \(p(x)\) to find the remainder \(r\):

\(x-1=0  → x=1\)

\(p(1)=3(1)^5-(1)^4+(1)^3-4(1)^2+2\)

\(=3-1+1-4+2\)

\(=1\)

Therefore, the remainder is \(1\).

The Remainder Theorem, Example 2:

Find the remainder when \(p(x)=x^3-x^2+x-1\) is divided by \(q(x):x+1\).

Replace the zero of \(q(x)\) into the polynomial \(p(x)\) to find the remainder \(r\):

\(x+1=0 → x=-1\)

\(p(-1)=(-1)^3-(1)^2+(-1)-1\)

\(=-1-1-1-1\)

\(=-4\)

Therefore, the remainder is \(-4\).

Exercises for the Remainder Theorem

1. Find the remainder after \(2x^2-5x-1\) is divided by \(x-5\).
2. Use the remainder theorem to evaluate \(f(x)=2x^5+4x^4-3x^3+8x^2+7\) at \(x=2\).
3. Find the remainder when \(4x^3-5x+1\) is divided by \(2x-1\).
4. Use the remainder theorem to find the remainder \((x^6+4x^5+9x^3-4x^2+10) \div (x+1)\).

1. \(\color{blue}{24}\)
2. \(\color{blue}{143}\)
3. \(\color{blue}{-1}\)
4. \(\color{blue}{-6}\)

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