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Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara

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Let's think step by step: 1) For x2 + 2x + 5 to be a factor of x4 + px2 + q, when we divide x4 + px2 + q by x2 + 2x + 5, the remainder should be 0. 2) Dividing x4 + px2 + q by x2 + 2x + 5, the quotient will be x2 and the remainder will be p - 5. 3) For the remainder to be 0, we must have p - 5 = 0. Therefore, p = 5 4) Putting p = 5 in the original expression, we get: x4 + 5x2 + q 5) Comparing the constant

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POLYNOMIALS FOR CLASS 10 PREPARED BY GOLAM ROBBANI AHMED
a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer as exponents of variables. An example of a polynomial of a single variable x is x2 − 4x + 7. An example of Polynomial in three variables is x3 + 2xyz2 − yz + 1
What is Variable? A variable is a quantity that may change within the context of a mathematical problem or experiment. Typically, we use a single letter to represent a variable. The letters x, y, and z are common generic symbols used for variables. Sometimes, we will choose a letter that reminds us of the quantity it represents, such as t for time, v for voltage etc.
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Based on the number of terms of the given polynomial, it can be divided into monomial, binomial, trinomial, constant polynomials. The polynomial with only one term is called monomial. The polynomial with two terms is called binomial. The polynomial with three terms are called trinomial.
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Based on the non negative integer exponent of Variable
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial Degree Name using Degree Number of Terms Name using number of terms 7x + 4 1 Linear 2 Binomial 3x2 + 2x + 1 2 Quadratic 3 Trinomial 4x3 3 Cubic 1 Monomial 9x4 + 11x 4 Fourth degree 2 Binomial 5 0 Constant 1 monomial
FOLLOWINGS ARE NOT POLYNOMIAL
Standard form of Polynomial: The polynomial above is in standard form. Standard form of a polynomial - means that the degrees of its monomial terms decrease from left to right.
The degree of a polynomial: The degree of a polynomial in one variable is the largest exponent in the polynomial. Ex: 5x12−2x6+x5−198x+1 degree :12 X4−x3+x2−x+1 degree :4 -8 degree: 0 On the otherhand Polynomials in two variables are algebraic expressions consisting of terms in the form axnym The degree of each term in a polynomial in two variables is the sum of the exponents in each term and the degree is the highest sum.
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Relationship between the zeros and coefficients of a quadratic polynomial?
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Q. If the squared difference of the zeros of the quadratic polynomial x² + px + 45 is equal to 144 , find the value of p. Ans: Let two zeros are α and β where α > β According given condition (α - β)2 = 144 Let p(x) = x² + px + 45 α + β = − p αβ = 45 now (α + β)² = (α - β)² + 4 αβ (-P)² = 144 +180=324 Solving this we get p = ± 18
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Question : If two zeroes of the polynomial are find other zeros = x² + 4 − 4x − 3 = x² − 4x + 1 is a factor of the given polynomial For finding the remaining zeroes of the given polynomial, we will find the quotient by dividing
= = Or x = 7 or −5 Hence, 7 and −5 are also zeroes of this polynomial.
Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara
Q. if p(x) a Polynomial and p(a)p(b)<0 then find number of zeroes of the polynomial between a and b. Here p(a)p(b)<0 i.e. if P(a) is positive then P(b) is negative and vice versa. This will be possible when P(x) is either linear, cubic,and so on i.e zeroes will be 1,3,5,………………….
Q.What will be the value of p and q for x2 + 2x + 5 to be a factor of x4 + px2 + q

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Polynomial for class 10 by G R Ahmed TGT (Maths) at K V Khanapara

  • 1. POLYNOMIALS FOR CLASS 10 PREPARED BY GOLAM ROBBANI AHMED
  • 2. a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer as exponents of variables. An example of a polynomial of a single variable x is x2 − 4x + 7. An example of Polynomial in three variables is x3 + 2xyz2 − yz + 1
  • 3. What is Variable? A variable is a quantity that may change within the context of a mathematical problem or experiment. Typically, we use a single letter to represent a variable. The letters x, y, and z are common generic symbols used for variables. Sometimes, we will choose a letter that reminds us of the quantity it represents, such as t for time, v for voltage etc.
  • 5. Based on the number of terms of the given polynomial, it can be divided into monomial, binomial, trinomial, constant polynomials. The polynomial with only one term is called monomial. The polynomial with two terms is called binomial. The polynomial with three terms are called trinomial.
  • 7. Based on the non negative integer exponent of Variable
  • 9. Polynomial Degree Name using Degree Number of Terms Name using number of terms 7x + 4 1 Linear 2 Binomial 3x2 + 2x + 1 2 Quadratic 3 Trinomial 4x3 3 Cubic 1 Monomial 9x4 + 11x 4 Fourth degree 2 Binomial 5 0 Constant 1 monomial
  • 10. FOLLOWINGS ARE NOT POLYNOMIAL
  • 11. Standard form of Polynomial: The polynomial above is in standard form. Standard form of a polynomial - means that the degrees of its monomial terms decrease from left to right.
  • 12. The degree of a polynomial: The degree of a polynomial in one variable is the largest exponent in the polynomial. Ex: 5x12−2x6+x5−198x+1 degree :12 X4−x3+x2−x+1 degree :4 -8 degree: 0 On the otherhand Polynomials in two variables are algebraic expressions consisting of terms in the form axnym The degree of each term in a polynomial in two variables is the sum of the exponents in each term and the degree is the highest sum.
  • 17. Relationship between the zeros and coefficients of a quadratic polynomial?
  • 20. Q. If the squared difference of the zeros of the quadratic polynomial x² + px + 45 is equal to 144 , find the value of p. Ans: Let two zeros are α and β where α > β According given condition (α - β)2 = 144 Let p(x) = x² + px + 45 α + β = − p αβ = 45 now (α + β)² = (α - β)² + 4 αβ (-P)² = 144 +180=324 Solving this we get p = ± 18
  • 22. Question : If two zeroes of the polynomial are find other zeros = x² + 4 − 4x − 3 = x² − 4x + 1 is a factor of the given polynomial For finding the remaining zeroes of the given polynomial, we will find the quotient by dividing
  • 23. = = Or x = 7 or −5 Hence, 7 and −5 are also zeroes of this polynomial.
  • 25. Q. if p(x) a Polynomial and p(a)p(b)<0 then find number of zeroes of the polynomial between a and b. Here p(a)p(b)<0 i.e. if P(a) is positive then P(b) is negative and vice versa. This will be possible when P(x) is either linear, cubic,and so on i.e zeroes will be 1,3,5,………………….
  • 26. Q.What will be the value of p and q for x2 + 2x + 5 to be a factor of x4 + px2 + q