More about Polynomials

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More about Polynomials
Unit 3
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3.1 Review on Polynomials
(A) Monomials and Polynomials
A monomial is a an algebraic expression
containing one term, which may be a constant, a
positive integral power of a variable or a
product of powers of variables. e.g. 4, 2x3 and
3x2y
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Polynomials

• A polynomials contains one terms or a sum of
  terms.

Each term of a polynomial is a product of a
constant (coefficient) and one or more variables
whose exponents are non-negative integers.
e.g. 6a3, 4x3 x, 3y4 2y2 1, 6x2y2 xy y
-ve e.g.
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The degree of a polynomial is equal to the
highest degree of its terms.
The terms of a polynomials are usually written in
descending order (i.e. the terms are arranged in
descending degree).
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Terms to Remember
i.e. y
3y4 2y2 1
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Terms to Remember
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Terms to Remember
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quotient
divisor
dividend
remainder
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Equality of Polynomials
If two polynomials in x are equal for all values
of x, then the two polynomials are identical, and
the coefficients of like powers of x in the two
polynomials must be equal.
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Alternative Method
When x 2, 3(2)2 - 5(2) - 5 A3(2)(2-2) B
12-10-5 B B
-3
When x 0, 3(0)2 - 5(0) 5 A3(0)(0-2) B
-5 -2A B
-5 -2A 3 -2 -2A
A 1
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(B) Remainder Theorem
remainder
28 3 x 9 1
quotient
dividend
divisor
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(c) Factor Theorem
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Corollary of Factor Theorem
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Applications of Theorems about Polynomials

• Use Factor Theorem to factorize a
• polynomial of degree 3 or above

• try to put a 1, -1, 2, -2, 3, -3, . one by
  one into the polynomial until the function is
  equal to zero.
• as the function is equal to zero, then (x a) is
  one of the factors.
• divide the polynomial by (x a) to get the
  quotient which is the other factor of the
  polynomial.
• factorize the quotient by the method you have
  learnt in before.

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Harder Algebraic Fractions

• (A) Find H.C.F. and L.C.M. of Polynomials

The HCF of two or more polynomials is the
polynomial of highest degree which is common
factor of the given polynomials.
e.g. f(x) a3 7a 6 (a 1)(a 2)(a
3) g(x) a4 1 (a2 1)(a 1)(a
1)
HCF a - 1
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The LCM is the polynomial of lowest degree which
is exactly divisible by each of the given
polynomials.
e.g. f(x) a3 7a 6 (a 1)(a 2)(a
3) g(x) a4 1 (a2 1)(a 1)(a
1)
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Simplification of Harder Algebraic Fractions

• (2) Signs of Fractions
• (i)
• (ii)
• (iii)