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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