General form of a polynomial in \(x\)

Algebraic expressions such as \(√x + x + 5, ~x^2 + \frac{1}{x^2}\)

Degree of a polynomial is highest power of the variable \(x\)

- Polynomial of degree 1 is known as linear polynomial.

Standard form is \(ax + b\), where \(a\) and \(b\) are real numbers and \(a≠0\) .

\(2x + 3\)is a linear polynomial. - Polynomial of degree 2 is known as quadratic polynomial.

Standard form is \(ax^2 + bx + c\), where \(a,~ b ~and ~c\) are real numbers and \(a ≠ 0\)

\(x^2 + 3x + 4\)is an example for quadratic polynomial. - Polynomial of degree 3 is known as cubic polynomial.

Standard form is \(ax^3 + bx^2 + cx + d\), where \(a,~ b,~ c ~and~ d\) are real numbers and \(a≠0\) .

\(x^3 + 4x + 2\)is an example for cubic polynomial.

Similarly,

\(y^6 + 3y^4 + y\)

Example: What is the value of a if degree of polynomial \(x^3 + x^{a-4} + x^2 + 1\)

Degree of a polynomial \(P(x)\)

Therefore, \(x^3 + x^{a-4} + x^2 + 1\)

\(a-4\)

Consider, \(P(x)\)

Put \(x\)

\(P(3)\)

Replace \(x\)

Similarly, value of \(x^2 – 3x + 2\)

\(P(0)\)

In general; if P(x) is a polynomial in x and k is any real number, then value of \(P(k)\)

In the polynomial \(x^2 – 3x + 2\)

Replacing \(x\)

\(P(1)\)

Similarly, replacing \(x\)

\(P(2)\)

For a polynomial \(P(x)\)

\(P(k)\)

Therefore, 1 and 2 are the zeros of polynomial \(x^2 – 3x + 2\)

Consider, \(P(x)\)

Let \(a\)

\(P(a)\)

Therefore,\(~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\)

In general, If \(k\)

\(P(k)\)

\(k\)

It can also be written as,

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