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**## Summary of the Topic

**Remainder Theorem**

If a polynomial \(p\left( x \right)\) is divided by a linear divisor \(x-a\), then the remainder is \(p\left( a \right)\).

**To find Remainder (without dividing) when a polynomial is divided by a Linear Polynomial**

**Example 1:** Find the remainder when \(9{{x}^{2}}-6x+2\) is divided by \(x-3\) and \(3x+1\)

**Example 2:** Find the value of \(k\) if the expression \({{x}^{3}}+k{{x}^{2}}+3x-4\) leaves a remainder of -2 when divided by \(x+2\).

**Zero of a Polynomial**

If a specific number \(x=a\) is substituted for the variable \(x\) in a polynomial \(p\left( x \right)\) so that the value \(p\left( a \right)\) is zero, then \(x=a\) is called zero of the polynomial.

**Factor Theorem**

The factor theorem helps us to find factors of polynomials because it determines whether a given linear polynomial \(x-a\) is a factor of \(p\left( x \right)\). All we need is to check whether \(p\left( a \right)=0\).

**Example 1:** Determine \(\left( x-2 \right)\) is a factor of \({{x}^{3}}-4{{x}^{2}}+3x+2\)

**Example 2:** Find a polynomial \(p\left( x \right)\) of degree 3 that has 2, -1, and 3 as zeros (roots)

**Factorizing of a Cubic Polynomial**

We can use factor theorem by the following steps

- Find a factor of the given polynomial by checking if \(x=a\) , then \(p\left( a \right)=0\) or not.
- Divide the cubic polynomial by \(p\left( x \right)\) to find the second factor which will be quadratic polynomial
- Factorize the quadratic polynomial.

## Videos

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## Test Yourself

You will have a test with 20 multiple choice questions.