Understand Polynomial. click on the Link to Watch the VIDEO explanation: Watch Video
Polynomial
A Polynomial is an algebraic expression that can have one or more variables.
Example
5x cube plus 3x square plus 2x plus 1
This is a polynomial in one variables x of degree 3.
A general form for a polynomial in x of degree n is:
P of x is equal to a not x power of n plus a1 x power of n minus 1 plus a2 x power n minus 2 and so on plus an minus 1 plus an
Where a not a1 a2 and so on upto an minus 1 and an are real number and an is not equal to 0 (because if a not is equal to 0 then there will not be any term of degree n)
Value of a Polynomial
The value of a polynomial p of x for x is equal to a is obtained by substituting x is equal to a in the given polynomial and it is denoted by p of a.
Example
P of x is equal to 3x square minus 7x minus 1,find p of 5 and p of minus 1 by 3.
Suggested answer:
P of 5 is equal to 3 into 5 square minus 7 of 5 minus 1, here a is equal to 5 that is equal to 39
P of minus 1 by 3 is equal to 3 into minus 1 by 3 square minus 7 into minus 1 by 3 – 1, here a is equal to minus 1 by 3 that is equal to 1 2by3.
Division of Polynomials
Consider a polynomial p of x.
Let us divide it by another polynomial d of x.
Let the quotient be q of x and the remainder be r of x.
Then p of x is equal to d of x into q of x plus r of x.
R of x is either is equal to 0 or the degree of r of x is less than the degree of d of x.
An example is solved for better understanding go through.
Note – observe that the degree of r of x is 0, since it is a constant equal to 17 and the degree of d of x is 1, since the power of x in x plus 2 is 1.
Remainder Theorem
If a polynomial f of x is divide by x minus a, then the remainder is f of a.
Examples
If f of x is divided by x minus 2 then the remainder is f of 2
If f of x is divided by x plus 2 then the remainder is f of minus 2
If f of x is divided by 2x minus 1 then the remainder is f of 1 by 2
Consider f of x is equal to 5x cube minus 2x square plus 7x minus 3.
In example 1, the divisor is x minus 2, remainder is f of 2.
F of 2 is equal to 5 into 2 cube minus 2 into 2 square plus 7 into 2 minus 3 that is 40 minus 8 plus 14 minus 3 that is equal to 43
Note – The value of the remainder can be verified by actual division of 5x cube minus 2x square plus 7x minus 3 by x minus 2.
In example 2, the divisor is x plus 2 remainder is f of minus 2.
F of x is equal to 5 x cube minus 2 x square plus 7 plus minus 3
F of minus 2 is equal to 5 of minus 2 cube minus 2 into minus 2 square plus 7 into minus 2 minus 3
i.e. equal to minus 40 minus 8 minus 14 minus 3
i.e. equal to minus 65
The value of the remainder can be verified by actual division of 5 x cube minus 2 x square plus 7 x minus 3 by x plus2.
Example 3, divisor is 2x minus 1, remainder is f of 1 by 2.
F of x is equal to 5 x cube minus 2 x square plus 7 plus minus 3
F of 1 by 2 is equal to 5 of 1 by 2 the whole cube minus 2 of I by 2 whole square plus 7 of 1 by 2 minus 3
i.e. equal to 5 by8 minus 2 by 4 plus 7 by 2 minus 3
i.e. equal to 5 by 8
Note The value of the remainder can be verified by actual division of 5 x cube minus 2 x square plus 7 x minus 3 by x plus2.
Factor Theorem
If a polynomial f of x is divided by x minus a and if the remainder f of a is equal to 0, then x minus a Is a factor of x.
Examples
1. Let f of x is equal to x square minus 7 x plus 6, be divided by x minus 1.
Then, remainder is equal to f of 1 by remainder theorem.
F of x is equal to x square minus 7x plus 6
Therefore, f of 1 is equal to 1 square minus 7 into 1 plus 6
Therefore f of 1 is equal to 0.
Implies x minus 1 is a factor of x square minus 7x plus 6.
Note
Verify by factorizing x square minus 7x plus 6.
X square minus 7x plus 6 is equal to x minus 1 into x minus 6
Therefore, x minus 1 is a factor of x square minus 7x plus 6.
2. Let f of x is equal to 2 x cube plus x square minus 5x plus 2 , be divided by x plus 2.
Then, remainder is equal to f of minus 2.
F of x is equal to 2 x cube plus x square minus 5x plus 2.
Therefore, f of minus 2 is equal to 2 into minus 2 cube plus minus 2 square minus 5 into minus 2 plus 2. That is equal to minus 16 plus 4 plus 10 plus 2.
F of minus 2 is equal to 0.
Implies x plus 2 is a factor of 2 x cube plus x square minus 5x plus 2.
Note
Verify by actual division.
Polynomial
A Polynomial is an algebraic expression that can have one or more variables.
Example
5x cube plus 3x square plus 2x plus 1
This is a polynomial in one variables x of degree 3.
A general form for a polynomial in x of degree n is:
P of x is equal to a not x power of n plus a1 x power of n minus 1 plus a2 x power n minus 2 and so on plus an minus 1 plus an
Where a not a1 a2 and so on upto an minus 1 and an are real number and an is not equal to 0 (because if a not is equal to 0 then there will not be any term of degree n)
Value of a Polynomial
The value of a polynomial p of x for x is equal to a is obtained by substituting x is equal to a in the given polynomial and it is denoted by p of a.
Example
P of x is equal to 3x square minus 7x minus 1,find p of 5 and p of minus 1 by 3.
Suggested answer:
P of 5 is equal to 3 into 5 square minus 7 of 5 minus 1, here a is equal to 5 that is equal to 39
P of minus 1 by 3 is equal to 3 into minus 1 by 3 square minus 7 into minus 1 by 3 – 1, here a is equal to minus 1 by 3 that is equal to 1 2by3.
Division of Polynomials
Consider a polynomial p of x.
Let us divide it by another polynomial d of x.
Let the quotient be q of x and the remainder be r of x.
Then p of x is equal to d of x into q of x plus r of x.
R of x is either is equal to 0 or the degree of r of x is less than the degree of d of x.
An example is solved for better understanding go through.
Note – observe that the degree of r of x is 0, since it is a constant equal to 17 and the degree of d of x is 1, since the power of x in x plus 2 is 1.
Remainder Theorem
If a polynomial f of x is divide by x minus a, then the remainder is f of a.
Examples
If f of x is divided by x minus 2 then the remainder is f of 2
If f of x is divided by x plus 2 then the remainder is f of minus 2
If f of x is divided by 2x minus 1 then the remainder is f of 1 by 2
Consider f of x is equal to 5x cube minus 2x square plus 7x minus 3.
In example 1, the divisor is x minus 2, remainder is f of 2.
F of 2 is equal to 5 into 2 cube minus 2 into 2 square plus 7 into 2 minus 3 that is 40 minus 8 plus 14 minus 3 that is equal to 43
Note – The value of the remainder can be verified by actual division of 5x cube minus 2x square plus 7x minus 3 by x minus 2.
In example 2, the divisor is x plus 2 remainder is f of minus 2.
F of x is equal to 5 x cube minus 2 x square plus 7 plus minus 3
F of minus 2 is equal to 5 of minus 2 cube minus 2 into minus 2 square plus 7 into minus 2 minus 3
i.e. equal to minus 40 minus 8 minus 14 minus 3
i.e. equal to minus 65
The value of the remainder can be verified by actual division of 5 x cube minus 2 x square plus 7 x minus 3 by x plus2.
Example 3, divisor is 2x minus 1, remainder is f of 1 by 2.
F of x is equal to 5 x cube minus 2 x square plus 7 plus minus 3
F of 1 by 2 is equal to 5 of 1 by 2 the whole cube minus 2 of I by 2 whole square plus 7 of 1 by 2 minus 3
i.e. equal to 5 by8 minus 2 by 4 plus 7 by 2 minus 3
i.e. equal to 5 by 8
Note The value of the remainder can be verified by actual division of 5 x cube minus 2 x square plus 7 x minus 3 by x plus2.
Factor Theorem
If a polynomial f of x is divided by x minus a and if the remainder f of a is equal to 0, then x minus a Is a factor of x.
Examples
1. Let f of x is equal to x square minus 7 x plus 6, be divided by x minus 1.
Then, remainder is equal to f of 1 by remainder theorem.
F of x is equal to x square minus 7x plus 6
Therefore, f of 1 is equal to 1 square minus 7 into 1 plus 6
Therefore f of 1 is equal to 0.
Implies x minus 1 is a factor of x square minus 7x plus 6.
Note
Verify by factorizing x square minus 7x plus 6.
X square minus 7x plus 6 is equal to x minus 1 into x minus 6
Therefore, x minus 1 is a factor of x square minus 7x plus 6.
2. Let f of x is equal to 2 x cube plus x square minus 5x plus 2 , be divided by x plus 2.
Then, remainder is equal to f of minus 2.
F of x is equal to 2 x cube plus x square minus 5x plus 2.
Therefore, f of minus 2 is equal to 2 into minus 2 cube plus minus 2 square minus 5 into minus 2 plus 2. That is equal to minus 16 plus 4 plus 10 plus 2.
F of minus 2 is equal to 0.
Implies x plus 2 is a factor of 2 x cube plus x square minus 5x plus 2.
Note
Verify by actual division.
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