There we discussed about Remainder Theorem, Factor Theorem and explained the Method of factoring with Example.

Here, we will apply the method to solve problems. The knowledge of Synthetic Division is made use of here. That knowledge is a prerequisite here. So, please learn the method before proceeding further.

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Solve the following problem on Polynomial Factoring

Factorize 3x^{4} - 10x^{3} + 5x^{2} + 10x - 8

Solution to Example 1 of Polynomial Factoring

Let f(x) = 3x^{4} - 10x^{3} + 5x^{2} + 10x - 8 Here constant term = -8. Its factors are +1,-1,+2, -2, +4, -4, +8, -8 So we check witha = +1,-1,+2, -2, +4, -4, +8, -8, whether f(a) is zero or not. f(1) = 3 - 10 + 5 + 10 - 8 = 0 < BR> ⇒ x - (1) = (x - 1) is a factor of f(x). Now let us divide f(x) = (3x^{4} - 10x^{3} + 5x^{2} + 10x - 8) by (x - 1) usingSynthetic Division (For detailed explanation of Synthetic Division see Example 1 above.)

Let p(x) = 3x^{3} - 7x^{2} - 2x + 8This is a third Degree Polynomial, which is to be factorized using Facor Theorem. Here constant term = -8. Its factors are +1,-1,+2, -2, +4, -4, +8, -8 So we check witha = +1,-1,+2, -2, +4, -4, +8, -8, whether f(a) is zero or not. p(1) = 3 - 7 -2 + 8 = 2 ≠ 0 p(-1) = 3(-1)^{3} - 7(-1)^{2} -2(-1) + 8 = -3 - 7 + 2 + 8 = 0. ⇒ x -(-1) = (x + 1) is a factor of p(x). Now let us divide p(x) = (3x^{3} - 7x^{2} - 2x + 8) by (x + 1) usingSynthetic Division (For detailed explanation of Synthetic Division see Example 1 above.)

Let f(x) = x^{4} + 2x^{3} - 8x^{2} + 30x - 25 Here constant term = -25. Its factors are +1,-1,+5, -5, +25, -25, So we check witha = +1,-1,+5, -5, +25, -25 whether f(a) is zero or not. f(1) = 1 + 2 - 8 + 30 - 25 = 0 < BR> ⇒ x - (1) = (x - 1) is a factor of f(x). Now let us divide f(x) = (x^{4} + 2x^{3} - 8x^{2} + 30x - 25) by (x - 1) usingSynthetic Division (For detailed explanation of Synthetic Division see Example 1 above.)

Let p(x) = x^{3} + 3x^{2} - 5x + 25This is a third Degree Polynomial, which is to be factorized using Facor Theorem. Here constant term = +25. Its factors are +1,-1,+5, -5. So we check witha = +1,-1,+5, -5, whether f(a) is zero or not. p(1) = 1 + 3 -5 + 25 = 24 ≠ 0 p(-1) = (-1)^{3} + 3(-1)^{2} -5(-1) + 25 = 32 ≠ 0. p(5) = 5^{3} + 3(5)^{2} - 5(5) + 25 ≠ 0 p(-5) = (-5)^{3} + 3(-5)^{2} - 5(-5) + 25 = -125 + 75 + 25 +25 = 0 ⇒ x -(-5) = (x + 5) is a factor of p(x). Now let us divide p(x) = (x^{3} + 3x^{2} - 5x + 25) by (x + 5) usingSynthetic Division (For detailed explanation of Synthetic Division see Example 1 above.)

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Solve the Following Problems on Polynomial Factoring :

Factorize 2x^{4} - 5x^{2} + 5x - 2

Factorize x^{4} + 4x^{3} + 3x^{2} - 4x - 4

For Answers of these problems on Polynomial Factoring See at the bottom of the Page.

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