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what is a polynomial

Variables, Constants, Coefficients of Variables and non-debt An algebraic expression with the operation of addition, subtraction or multiplication of an exponent is called a polynomial.

Example:

x2+4x−7×2+4x-7, x3+2x2y−y+1×3+2x2y-y+1, 3x3x, 5, etc.

On the other hand x−2yx-2y, 1x1x, 2x+12x+1, xx, etc. are not polynomials (POLYNOMIAL). Because A polynomial cannot contain the following expressions, or is not called a polynomial with the following expressions:

(i) Exponents with negative sign such as −2-2, −5-5, etc.

what is a polynomial

(ii) any term divisible by a variable, such as 1x1x

(iii) any exponent with a fraction such as xx, as it is written as x12x12.

But a polynomial can have constants, variables or powers.

Example

Constants: 3,2,−2,143,2,-2,14 etc.

Variables: x,yx,z,abcx,yx,z,abc, etc.

Exponents: 0,1,2,3,40,1,2,3,4, etc.

1-Power of Polynomial

If p(x)p(x) is a polynomial (POLYNOMIAL), then the highest power of xx in the polynomial p(x)p(x) of variable xx is called the degree of the polynomial.

2-linear polynomial

Let 4x+24x+2 be a polynomial.

The degree of the variable xx of this polynomial is one (1). Therefore, this polynomial is called a degree polynomial or an exponential polynomial or a linear polynomial.

Therefore, a polynomial of degree 1 is called a polynomial of degree 1 or a linear polynomial.

3-quadratic polynomial

Let x2+x+2×2+x+2 be a polynomial.

In this polynomial, the highest degree of the variable xx is 2 (two). Therefore, such a polynomial is called a quadratic polynomial or a quadratic polynomial.

Thus, the polynomial of degree 2 (two) is Quadratic Polynomial (QUADRATIC POLYNOMIAL) it is said.

what is a polynomial

4-quadratic polynomial

Polynomial to the Power of 3 (Three) Quadratic Polynomial (CUBIC POLYNOMIAL) It is called

Example:

x3+2×2−x+1×3+2×2-x+1, 2−x32-x3, 2x2x, etc.

Since the highest degree of the variable xx in these polynomials is 3 (three), all these Cubic Polynomial Huh.

Cubic Polynomial The most widespread form is:

ax3+bx2+cx+dax3+bx2+cx+d where a,b,c,da,b,c,d are real numbers and a≠0a≠0.

5-value of the polynomial

If p(x)p(x) is any polynomial in xx and kk is any real number, the real number p(x)p(x) obtained by substituting xx in p(x)p(x) by kk is is called the value at x=kx=k and is denoted by p(k)p(k).

Example:

Let p(x)=x2−3x−4p(x)=x2-3x-4

Substituting x=2x=2 in this, we get that

p(2)=223×2−4=−6p(2)=223×2-4=-6

The value 22 obtained here is called the x=2x=2 value of p(x)p(x).

what is a polynomial

6-zero of the polynomial

a real number kk polynomial Zero of a polynomial is called, if p(k)=0p(k)=0.

Example:

Let a polynomial p(x)=x2−3x−4p(x)=x2-3x-4

Substituting x=−1x=-1 in this polynomial, we get

p(−1)=(−1)2−3(−1)−4p(-1)=(-1)2-3(-1)-4

p(−1)=1+3−4=0⇒p(-1)=1+3-4=0

Now on substituting x=4x=4 in this polynomial, we get

p(4)=42?(3×4)−4p(4)=42?(3×4)-4

p(4)=16−12−4=0⇒p(4)=16-12-4=0

Since here p(−1)=0p(-1)=0 and p(4)=0p(4)=0

Hence −1-1 and 44 are called zeroes of the given polynomial x2−3x−4×2-3x-4 .

Zero of 7-Linear Polynomial

If kk is the zero of the polynomial p(x)=ax+bp(x)=ax+b, then

p(k)=ak+b=0p(k)=ak+b=0

ie k=−bak=-ba

Thus, the zero of the given linear polynomial (LINEAR POLYNOMIAL) ax+bax+b is equal to −ba-ba.

In general, if kk is a zero of p(x)=ax+bp(x)=ax+b, then p(k)=ak+b=0p(k)=ak+b=0

That means k=−bak=-ba.

Thus, the zero of the linear polynomial ax+bax+b

Thus, the zeroes of the linear polynomial are related to its coefficients. (Thus, zero of a LINEAR POLYNOMIAL is related to its coefficients.)

8- Geometrical meaning of the zeroes of the polynomial

In general, for a given polynomial p(x)p(x) of degree nn, the graph of y=p(x)y=p(x) intersects the x–x-axis at at most nn points . Thus, any polynomial of degree nn can have at most nn zeroes.

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