Question

Identifying factors of polynomials using the remainder theorem explain?

x^3+ 5x^2 +2x-8

Answer 1

Step 1:

Remainder theorem:

When we divide a polynomial by the remainder equals .

The polynomial

Rational Root Theorem : If a rational number in simplest form p/q is a root of the polynomial equation a_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀ = 0, where p is a factor of a₀ and q is a factor if a_n.

If p /q is a rational zero, then p is a factor of 8 and q is a factor of 1.

The possible values of p are  ± 1, ± 2, ± 4 and  ± 8.

The possible values for q are ± 1.

So, p/q = ± 1, ± 2, ± 4 and  ± 8.

Make a table for the synthetic division and test possible  zeros.

p/q	1	5	2	-8
-1	1	4	-2	-6
1	1	6	8	0

From remainder theorem:

is not a factor of .

is a factor of .

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Step 2:

Since , is a zero. The depressed polynomial is  .

Since the depressed polynomial of this zero, , is quadratic, use the Factorization method to find the roots of the related quadratic equation .

and

and

,   are also factors of .

Factored form of .

Solution:

Factored form of .