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10
UNIT
2
Polynomials. Algebraic Fractions
2.4
2.1
2.3
DIVISION OF POLYNOMIALS
To divide a monomial by a monomial, to divide the numerical coefficients and then subtract the exponents of the same variables.
EXAMPLES
Ruffini
(3x3+13x2-13x+2): (x-1)
Division of polynomials
EXAMPLES
(3x⁴ + 5x³ - 2x + 3) / (x² - 3x + 2) = 3x² + 14x + 36 Remainder 78x - 69
3x⁴ - 9x³ + 6x²
------------------------------------
14x³ - 6x² - 2x + 3
14x³ - 42x² + 28x
----------------------------
36x² - 30x + 3
36x² - 108x + 72
-------------------
78x - 69
Ruffini's Rule's uses
If remainder = 0 the dividend is divisible by (x-a)
You can write that the polynomial is equal to the factor times the quotient.
2.2
- Theorem of remainder
IN THIS DIVISION
It is not divisible by (x+2)
Here we have another example of Ruffini's rule.
It is divisible by (x-2).
Furthermore, 2 is a divisor of -6.
EXERCISE
An exercise to prove your maths skills
Use the Remainder Theorem to determine if x = –4 is a solution of
f(x)= x⁶+5x⁵+5x⁴+5x³+2x²–10x–8
FACTORIZING POLYNOMIALS
Roots of a polynomial
The roots (or zeroes) of a polynomial are the solutions of the equation P(x) =0
Examples:
a) P(1) = 12 -1 = 0 P( -1) = (-1)2 - 1 = 0
b) The roots of P(x)= x2+4x-5; 5 and -1.
-One of the most important uses of Ruffini’s rule is to find the roots of a polynomial.
x4+x3-7x2-8x+12
-If the remainder of the division P(x) : (x-a) is 0, then the number “a” is a root of P(x).
-Fundamental Theorem of Algebra.
Factorization Techniques
Factorization techniques
1) Taking common factor:
Example: 8x²-2x = 2x·(4x-1)
2) Using polynomial identities:
-The square of the sum
-The square of the difference
-The product of a sum and a difference
3) Using the Fundamental Theorem of Algebra:
Example: -x²-x+6 = (-1)·(x-2)·(x+3)
4) Using Ruffini's Rule:
P(x) = x³-2x²-5x+6 = (x-1)·(x+2)·(x-3)
5) A combination of the previous one.
EXERCISES
2) Take out common factor or use the polynomial identities
a) x⁴+2x³+x²
b) 45x²-5x⁴
EXERCISE 1:
SOLUTIONS
SOLUTIONS
a) x²(x-1)(x-1)
b) 5x²(x-3)(x+3)
EXERCISE 2:
EXERCISES
1) Factorize these polynomials:
a) x³+2x²-5x-6
b) x⁴+2x³-23x²-60x
c) 9x⁴-36x³+26x²+4x-3
Solutions
a) (x+1) (x-2) (x+3)
b) x(x-5) (x+3) (x+4)
c) (x-3) (x-1) (3x-1) (3x+1)
ALGEBRAIC FRACTIONS
An algebraic fraction is the quotient of two polynomials, that is, P(x)/Q(x)
Simplification of algebraic fractions
To simplify algebraic fractions:
• Factorize the polynomials.
• Cancel out the common factors.
EXAMPLE
EXAMPLE
AND EXERCISES
EXERCISES
3x+5x-12x=
Link to Kahoot:
GAME
https://play.kahoot.it/v2/?quizId=40f9436d-11f8-4c3f-a256-be420d95397e