Find the square root of algebraic expression using division method 4x^4 -12x^3+25x^2-24x+16
Answers
Answer:
A polynomial is an algebraic expression of the type anxn + an−1xn−1+…………………a2x2 + a1x + a0, where “n” is either 0 or positive variables and real coefficients.
In this expression, an, an−1…..a1, a0 are coefficients of the terms of the polynomial.
The highest power of x in the above expression, i.e. n is known as the degree of the polynomial.
If p(x) represents a polynomial and x = k such that p(k) = 0 then k is the root of the given polynomial.
Given a polynomial equation, p(x)=x2–x–2. Find the zeroes of the equation.
Solution:
Given Polynomial, p(x)=x2–x–2
Zeroes of the equation is given by:
x2–2x+x–2=0
x(x−2)+1(x–2)
(x+1)(x−2)=0
⇒ x=−1
Or, x=2
Thus, -1 and 2 are zeroes of the given polynomial.
Example:
Given a polynomial equation, p(x)=x2–x–2. Find the zeroes of the equation.
Solution:
Given Polynomial, p(x)=x2–x–2
Zeroes of the equation is given by:
x2–2x+x–2=0
x(x−2)+1(x–2)
(x+1)(x−2)=0
⇒ x=−1
Or, x=2
Thus, -1 and 2 are zeroes of the given polynomial.
If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that
p(x) = g(x) × q(x) + r(x)
Here,
r(x) = 0 or degree of r(x) < degree of g(x)
This result is called the Division Algorithm for polynomials.
From the previous example, we can verify the polynomial division algorithm as:
p(x) = 3x3 + x2 + 2x + 5
g(x) = x2 + 2x + 1
Also, quotient = q(x) = 3x – 1
remainder = r(x) = 9x + 10
Now,
g(x) × q(x) + r(x) = (x2 + 2x + 1) × (3x – 5) + (9x + 10)
= 3x3 + 6x2 + 3x – 5x2 – 10x – 5 + 9x + 10
= 3x3 + x2 + 2x + 5
= p(x)
Hence, the division algorithm is verified.