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Chapters For Class X- CBSE


If p(x) is a polynomial in x, the highest power of x in p(x) is called degree of the polynomial p(x).
Examples: $4x+2$ is a polynomial in the variable x of degree 1.
4y2+5y+3 is a polynomial in the variable y of degree 2.
5z3+7z2+z+8 is a polynomial in the variable z of degree 3.

Note: Expression like $1/{(x+1)};1/{x^2+2x+3}; {√x+2 }$ are not polynomials.
Polynomial of degree 1 is called linear polynomial.(General form: $ax+b$) Example, $2x+3$
Polynomial of degree 2 is called quadratic polynomial.(General form: $ax^2+bx+c$) Example, $2x^2+3x-2/5$
Polynomial of degree 3 is called cubic polynomial. (General form: $ax^3+bx^2+cx+d$) Example, $5-u^3$ ,$√2x^3+x^2+x-8$



Now consider the polynomial $p(x) = x^2 – 3x – 4$. Then, putting $x = 2$ in the polynomial, we get $p(2) = 2^2 – 3 × 2 – 4 = – 6$.
The value ‘– 6’, obtained by replacing $x$ by $2$ in $x^2 – 3x – 4$, is the value of $x^2 – 3x – 4$ at x = 2.
Similarly, p(0) is the value of $p(x)$ at $x = 0$, which is – 4.

______________________________________________________________________________ verticle lines horizontals lines -5 -4 -3 -2 -1 1 2 3 4 5 x-axis 10 8 6 4 2 -2 -4 -6 -8 -10 (-2,6) (-1,0) (0,-4) (2,-6) (4,0)
If $p(x)$ is a polynomial in $x$, and if $k$ is any real number, then the value obtained by replacing $x$ by $k$ in $p(x)$, is called the value of $p(x)$ at $x = k$, and is denoted by $p(k)$.

What is the value of $p(x) = x^2 –3x – 4$ at $x = –1$? We have : $p(–1) = (–1)^2 –{3 × (–1)} – 4 = 0$ Also, note that $p(4) = 4^2 – (3 × 4) – 4$ = 0.
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p(x) -6-4600
As p(–1) = 0 and p(4) = 0, so, –1 and 4 are called the zeroes of the quadratic polynomial $x^2 – 3x – 4$ . More generally, a real number $k$ is said to be a zero of a polynomial $p(x)$, if p($k$) = 0.

A real number $k$ is said to be a zero of a polynomial $p(x)$, if p($k$) = 0.

How to find the zeroes of a linear polynomial. 

If $k$ is a zero of $p(x) = 2x + 3$, then $p(k) = 0$ gives us
$2k + 3 = 0$, i.e., $k$ = $ -{3}/{2} $
In general, if $k$ is a zero of $p(x) = ax + b$, then $p(k) = ak + b = 0$, i.e., k= $-{b/a}$ 

So, the zero of the linear polynomial $ax + b$ is  $-{b/a}$ =  $-{Constant term}/{Coefficient of x}$ 
Thus, the zero of a linear polynomial is related to its coefficients. 
 
So, zeroes of the cubic polynomial are $1,2 ,-2$