Chapter 2 Polynomials
2 Polynomials
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Fundamentals: •
An algebraic equation of the form, n n-1 n-2 p(x) = a0x + a1x + a2x + ……… + an-1x + an is called polynomial, provided it has no negative exponent for any variable. Where a0, a1, a2, ……an-1, an are constants (real numbers); a0 ≠ 0.
•
Remainder Theorem: 1. Remainder obtained on dividing polynomial p(x) by x – a is equal to p(a) . 2. If a polynomial p(x) is divided by (x + a) the remainder is the value of p(x) at x = –a. 3. (x – a) is a factor of polynomial p(x) if p (a) = 0 4. (x + a) is a factor of polynomial p(x) if p (–a) = 0 5. (x – a) (x – b) is a factor of polynomial p(x), if p(a) = 0 and p(b) = 0. Degree of polynomial: n is called the degree (highest power of variable x). If n = 1 then polynomial is called linear polynomial. General form: ax + b = 0 (a ≠ 0) If n = 2 then polynomial is called quadratic polynomial. General form: ax2 + bx + c = 0 (a ≠ 0) If n = 3 then polynomial is called cubic polynomial. General form: ax3 + bx2 + cx + d = 0 (a ≠ 0) Zeroes of Polynomial: For polynomial p(x), the value of x for which p(x) = 0, is called zero(es) of polynomial. Linear Polynomial can have at most 1 root. Quadratic Polynomial can have at most 2 roots. Cubic Polynomial can have at most 3 roots.
•
Types of Polynomials: (A) Based on degree: If degree of polynomial is
Examples
3x 3
1
One
Linear
2
Two
Quadratic
3
Three
Cubic
x3 + 3x2 – 7x + 8, 2x2 + 5x3 + 7
4
Four
bi-quadratic
x4 + y4 + 2x2y2, x4 + 3,…
x+3, y – x + 2, 2x2 – 7,
1 2 x + y2 2xy, x2 + 1 + 3y 3
(B) Based on Terms: If number of terms in polynomial is
Examples 7x, 5x9,
7 16 x , xy, ….. 3
1
One
Monomial
2
Two
Binomial
2 + 7y6, y3 + x14, 7 + 5x9,…
3
Three
Trinomial
x3 + 2x + y, x31 + y32 + z33,…
Relationship between zeroes and coefficient of polynomial: -b - ( cons tan t term) Zero of linear polynomial ax + b = 0 is given by x = a = coeffient of x . ( ) •
•
If a and ß are zeroes of the quadratic polynomial ax2 + bx + c = 0, then Sum of zeroes
a +b =
Products of zeroes
ab =
-b - ( coeffient of x ) = a coeffient of x 2
(
)
c ( cons tan t term) = a coeffient of x 2
(
)
If α, β and γ are the zeroes of the cubic polynomial ax + bx + cx + d = 0, then 3
2 -b - coeffient of x = a coeffient of x 3
Sum of zeroes
a +b+ g =
Product of zeroes
ab + bg + ga =
abg =
(
2
(
)
)
c - ( cons tanat of x ) = a coeffient of x 3
(
)
-d - ( cons tanat term) = a coeffient of x 3
(
)
Division algorithm for polynomials: If p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can find q(x) and r(x) such that p(x) = g(x) × q(x) + r(x) In simple words: Dividend = Divisor × Quotient + Remainder Tips: 1. Graph of linear polynomial is a straight line, while graph of quadratic equation is a parabola. 2. Degree of polynomial = maximum number of zeroes of polynomial. 3. If remainder r(x) = 0, then g(x) is a factor of p(x). 4. To form quadratic polynomial if sum and product of zeroes are given 2 p(x) = x – (Sum of zeroes) x + (Product of zeroes)
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Fundamentals: •
An algebraic equation of the form, n n-1 n-2 p(x) = a0x + a1x + a2x + ……… + an-1x + an is called polynomial, provided it has no negative exponent for any variable. Where a0, a1, a2, ……an-1, an are constants (real numbers); a0 ≠ 0.
•
Remainder Theorem: 1. Remainder obtained on dividing polynomial p(x) by x – a is equal to p(a) . 2. If a polynomial p(x) is divided by (x + a) the remainder is the value of p(x) at x = –a. 3. (x – a) is a factor of polynomial p(x) if p (a) = 0 4. (x + a) is a factor of polynomial p(x) if p (–a) = 0 5. (x – a) (x – b) is a factor of polynomial p(x), if p(a) = 0 and p(b) = 0. Degree of polynomial: n is called the degree (highest power of variable x). If n = 1 then polynomial is called linear polynomial. General form: ax + b = 0 (a ≠ 0) If n = 2 then polynomial is called quadratic polynomial. General form: ax2 + bx + c = 0 (a ≠ 0) If n = 3 then polynomial is called cubic polynomial. General form: ax3 + bx2 + cx + d = 0 (a ≠ 0) Zeroes of Polynomial: For polynomial p(x), the value of x for which p(x) = 0, is called zero(es) of polynomial. Linear Polynomial can have at most 1 root. Quadratic Polynomial can have at most 2 roots. Cubic Polynomial can have at most 3 roots.
•
Types of Polynomials: (A) Based on degree: If degree of polynomial is
Examples
3x 3
1
One
Linear
2
Two
Quadratic
3
Three
Cubic
x3 + 3x2 – 7x + 8, 2x2 + 5x3 + 7
4
Four
bi-quadratic
x4 + y4 + 2x2y2, x4 + 3,…
x+3, y – x + 2, 2x2 – 7,
1 2 x + y2 2xy, x2 + 1 + 3y 3
(B) Based on Terms: If number of terms in polynomial is
Examples 7x, 5x9,
7 16 x , xy, ….. 3
1
One
Monomial
2
Two
Binomial
2 + 7y6, y3 + x14, 7 + 5x9,…
3
Three
Trinomial
x3 + 2x + y, x31 + y32 + z33,…
Relationship between zeroes and coefficient of polynomial: -b - ( cons tan t term) Zero of linear polynomial ax + b = 0 is given by x = a = coeffient of x . ( ) •
•
If a and ß are zeroes of the quadratic polynomial ax2 + bx + c = 0, then Sum of zeroes
a +b =
Products of zeroes
ab =
-b - ( coeffient of x ) = a coeffient of x 2
(
)
c ( cons tan t term) = a coeffient of x 2
(
)
If α, β and γ are the zeroes of the cubic polynomial ax + bx + cx + d = 0, then 3
2 -b - coeffient of x = a coeffient of x 3
Sum of zeroes
a +b+ g =
Product of zeroes
ab + bg + ga =
abg =
(
2
(
)
)
c - ( cons tanat of x ) = a coeffient of x 3
(
)
-d - ( cons tanat term) = a coeffient of x 3
(
)
Division algorithm for polynomials: If p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can find q(x) and r(x) such that p(x) = g(x) × q(x) + r(x) In simple words: Dividend = Divisor × Quotient + Remainder Tips: 1. Graph of linear polynomial is a straight line, while graph of quadratic equation is a parabola. 2. Degree of polynomial = maximum number of zeroes of polynomial. 3. If remainder r(x) = 0, then g(x) is a factor of p(x). 4. To form quadratic polynomial if sum and product of zeroes are given 2 p(x) = x – (Sum of zeroes) x + (Product of zeroes)
