|Table of contents|
|PolynomialsDefinition of a Polynomial|
|General Theory of Equations|
Let be real numbers and x is a real variable. Then f(x) = is called a real polynomial of real variable x with coefficients
Example: is a polynomial
The degree of a polynomial is the highest power of the variable in the polynomial.
Q.1. Find the degree of the polynomial 6x4+ 3x2+ 5x +19
Ans: The degree of the polynomial is 4.
Q.2. Find the degree of the polynomial 6x3+4x3+3x+1
Ans: The degree of the polynomial is 3.
If any polynomial f(x) is divided by (x - a) then f(a) is the remainder.
If (x - a) is a factor of f(x), then remainder f(a) = 0. (Or) if f(a) = 0, then (x - a) is a factor of f(x),
Q.1. Find the remainder by using the remainder theorem when polynomial x3−3x2+x+1 is divided by x−1.
Q.2. Find the remainder when p(x)= −x+3x2−1 is divided by x+1.
A linear equation is 1st degree equation. It has only one root. Its general form is a
Quadratic Equation in “x” is one in which the highest power of “x” is 2. The equation is generally satisfied by two values of “x”.
The quadratic equation ax2 + bx + c = 0 will have maximum or minimum value at x = - b/2a. If a < 0, it has maximum value and if a > 0, it has minimum value.
The maximum or minimum value is given by
Ex.2. A and B went to a hotel paid Rs. 84 for 3 plates of Idli and 5 plates of Dosa. Whereas B took 5 plates of Idli and 3 plates of Dosa and paid Rs. 76. What is the cost of one plate of Idli.
- 3I + 5D = 84 ……….(1)
- 5I + 3D = 76 ………(2)
- Equation (1) × 3 - equation (2) × 5, we get
⇒ 16I = 128
⇒ I = 8
- Each plate of Idli cost Rs. 8.
Ex.3. Find the values of x and y from the equations
Ex.4. Find x and y from
Ex.5. Aman won a competition and so he got some prize money. He gave Rs. 2000 less than the half of prize money to his son and Rs. 1000 more than the two third of the remaining to his daughter. If both they got same amount, what is the prize money Aman got?
Ex.6. How many non negative integer pairs (x, y) satisfy the equation, 3x + 4y = 21?
- Since x and y are non negative integers. Start from x = 0.
- If x = 0 or 2, y cannot be integer.
- For x = 3, y = 3.
And for x = 7, y = 0.
- These two pairs only satisfy the given equation.
Ex.7. If (x - 2) is a factor of x3 - 3x2 + px + 4. Find the value of p.
- Since (x - 2) is a factor, f(2) = 0.
- ∴ 23 - 3(22) + (2)p + 4 = 0
⇒ p = 0
Ex.8. When x3 - 7x2 + 3x - P is divided by x + 3, the remainder is 4, then what is the value of P ?
- f(- 3) = 4
- ∴ (- 3)3 - 7(- 3)2 + 3(- 3) - P = 4
- P = - 103.
Ex.9. If (x - 1) is the HCF of (x3 - px2 + qx - 3) and (x3 - 2x2 + px + 2). What is the value of ‘q’?
- Since (x - 1) is HCF, it is a factor for both the polynomials.
- ∴ 13 - p(1)2 + q(1) - 3 = 0
- - p + q = 2
- And 13 - 2(12) + p(1) + 2 = 0
- p = - 1
- ∴ q = 1
Ex.10. Find the roots of the quadratic equation x2 - x - 12 = 0.
- x2 - x - 12 = 0
- x2 - 4x + 3x - 12 = 0
- x(x - 4) + 3(x - 4) = 0
- (x - 4) (x + 3) = 0
- x = 4 or -3
- The roots are 4 and -3.
|1. What is the definition of a polynomial?|
|2. What is the Remainder Theorem?|
|3. What is the Factor Theorem?|
|4. What is the General Theory of Equations?|
|5. What are linear equations and how are they different from quadratic equations?|