Friday, December 14

Polynomials Algebra Tiles


In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. In this article we shall discuss about polynomials algebra tiles. (Source: Wikipedia)


Sample Problem for Polynomials Algebra Tiles:

Solved polynomials algebra tiles problem:

Example 1:

Factorize the polynomial 2x3 – 10x2 – 24x+ 72

Solution

Sum of the coefficients of terms: 2–10–24 + 72 = 40 ≠ 0.

Therefore the value (x–1) is not a factor.

Sum of the coefficients of even degree terms = –10 + 72 = 62

Sum of the coefficients of odd degree terms = 2 – 24 = –22

Since they are not equivalent we estimate that (x + 1) is also not a factor. Let us check whether x – 2 is a factor. By synthetic division method

2 | 2       -10     -24        +72

|

|          +4      -12          -72
________________________

2        -6     -36     |    0

_________________________

Since the remainder is 0, (x – 2) is a factor. To find other factors

2x2 – 6x – 36 = 2x2 – 12 x + 6x –36

= 2x(x –6) + 6(x–6) = (2x + 6) (x – 6)

Therefore, x3 – 5x2 – 12x + 36 = (x–2) (x–6) (x+3)


Example 2:

Factorize 4x3 + 2x2 – 10x + 4

Solution:

Since the addition of the coefficients of all the terms: 4 + 2 – 10 + 4 = 10 – 10 = 0

We guess that (x – 1) is a factor.

By synthetic division,

1 | 4         +2        -10        +4

|

|              4        +6        -4

________________________

4         6         -4     |   0

_________________________

Remainder is 0. Quotient is 4x2 + 6x – 4

To find other factors, factorize the quotient,

4x2 + 6x – 4 = 4x2 + 8x – 2x – 4

= 4x (x + 2) – 2 (x + 2)

= (x + 2) (4x – 2)

∴ 2x3 + x2 – 5x + 2 = (x – 1) (x + 2) (2x – 1)
Practice Problem for Polynomials Algebra Tiles:

Factorize 2x2 + 3x -2

Answer: (x + 2) (2x – 1)

Factorize X2 – 3x – 18

Answer: (x + 3) (x – 6)

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