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Timeless Theorems of Mathematics/Polynomial Factor Theorem

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The Polynomial Factor Theorem is a theorem linking factors and zeros of a polynomial.[1] It is an application of the Polynomial Remainder Theorem. It states that a polynomial has a factor if and only if . Here, is also called the root of the polynomial.[2]

Proof

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Statement

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If is a polynomial of a positive degree and if so is a factor of .

Proof

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According to the Polynomial Remainder Theorem, the remainder of the division of by is equal to . As , so the polynomial is divisible by

is a factor of . [Proved]

Converse of Factor Theorem

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Proposition : If is a factor of the polynomial then

Factorization

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Example 1

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Problem : Resolve the polynomial into factors.

Solution : Here, the constant term of is and the set of the factors of is 1{±1, ±2}

Here, the leading coefficient of is and the set of the factors of is 2{±1, ±2, ±3, ±6, ±9, ±18}

Now consider , where 12

When,

Therefore, is a factor of

Now,

Example 2

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Problem : Resolve the polynomial into factors.

Solution : Considering only the terms of and constant, we get .

In the same way, considering only the terms of and constant, we get .

Combining factors of above (i) and (ii), the factors of the given polynomial can be found. But the constants must remain same in both equations just like the coefficients of and .


References

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  1. [1] Byjus.com, Maths, Factor Theorem
  2. [2] Byjus.com, Maths, Roots of Polynomials