For a polynomial, there could be some values of the variable for which the polynomial will be zero. These values are called zeros of a polynomial. Sometimes, they are also referred to as roots of the polynomials, In general, we use to find the zeros of quadratic equations, to get the solutions for the given equation.
The standard form of a polynomial in x is anxn + an-1xn-1 +….. + a1x + a0, where an, an-1, ….. , a1, a0 are constants, an ≠0 and n is a whole number. For example, algebraic expressions such as √x + x + 5, x2 + 1/x2 are not polynomials because all exponents of x in terms of the expressions are not whole numbers.
Finding Zeros of Polynomials
Zeros of a polynomial can be defined as the points where the polynomial becomes zero on the whole. A polynomial having value zero (0) is called zero polynomial. The degree of a polynomial is the highest power of the variable x.
- A polynomial of degree 1 is known as a linear polynomial.
The standard form is ax + b, where a and b are real numbers and a≠0.
2x + 3 is a linear polynomial.
- A polynomial of degree 2 is known as a quadratic polynomial.
Standard form is ax2 + bx + c, where a, b and c are real numbers and a ≠ 0
x2+ 3x + 4 is an example for quadratic polynomial.
- Polynomial of degree 3 is known as a cubic polynomial.
Standard form is ax3+ bxx2 + cx + d, where a, b, c and d are real numbers and a≠0.
x3 + 4x + 2 is an example for cubic polynomial.
y6 + 3y4 + y is a polynomial in y of degree 6.
Points to remember:
- 0 could be a zero of polynomial but it is not necessarily a zero has to be 0 only.
- All the linear polynomials have only one zero.
- The zeros of the polynomial depends on its degree.
Zeros of Polynomial Formula
Consider, P(x) = 4x + 5 to be a linear polynomial in one variable.
Let a be zero of P(x), then,
P(a) = 4k+5 = 0
Therefore, k = -5/4
In general, If k is zero of the linear polynomial in one variable; P(x) = ax +b, then
P(k) = ak+b = 0
k = -b/a
It can also be written as,
Example: What is the value of a if degree of polynomial x3 + xa-4 + x2 + 1 is 4?
Degree of a polynomial P(x) is the highest power of x in P(x).
Therefore, xa-4 = x4
a-4 = 4, a = 4+4 =8
Consider, P(x)= x2 – 3x + 2,
Put x = 3 in P(x) which gives,
P(3) = 9 – 9 + 2 = 2
Replace x by 2 in the polynomial x2 – 3x + 2, which gives P(2) = 4 – 6 + 2 = 0
Similarly, value of x2 – 3x + 2, at x = 0 is,
P(0) = 0-0+2 = 2
In general; if P(x) is a polynomial in x and k is any real number, then value of P(k) at x = k is denoted by P(k) is found by replacing x by k in P(x).
In the polynomial x2 – 3x + 2,
Replacing x by 1 gives,
P(1) = 1 – 3 + 2 = 0
Similarly, replacing x by 2 gives,
P(2) = 4-6+2 = 0
For a polynomial P(x), real number k is said to be zero of polynomial P(x), if P(k) = 0.
Therefore, 1 and 2 are the zeros of polynomial x2 – 3x + 2.
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