__10__^{th} Polynomials Ch-02-Mathematics [Key points] Check point [Formative Assessment]

Þ Any algebraic expression having non zero integral
power (whole number) is called polynomial.

Þ If p(x) is a polynomial in x, the highest power of x in p(x) is
called the degree of the polynomial p(x).

Þ A polynomial of degree 1 is called a linear polynomial. For
example, 2x – 3, √3x+1,y +√3

Þ A polynomial of degree 2 is called a quadratic polynomial.
The name ‘quadratic’ has been derived from the word ‘quadrate’, which means
‘square’. 2x^{2} + 3x
+ 2 ,y^{2} + 2

Þ Any quadratic polynomial in x is of the form ax^{2} +
bx + c, where a, b, c are real numbers and a ≠ 0.

Þ A polynomial of degree 3 is called a cubic polynomial e.g.
2 – *x*^{3}, *x*^{3}, √2 *x*^{3}.

Þ General form of a cubic polynomial is ax^{3} +
bx^{2} + c x + d, where, a, b, c, d are real numbers and a ≠ 0.

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

Þ Thus, the zero of a linear polynomial is related to its
coefficients because if *k *is a zero of *p*(*x*)
= *ax *+ *b*, then

*p *(*k*) = a
k + b = 0, i.e., k = -b/a

Þ Zero of a linear polynomial ax + b is −b/a

ÞThe graph of y = ax + b is a straight line like the graph of y =
ax + b is a straight line passing through the points (– 2, –1) and (2, 7) and
straight line straight line intersects the x-axis at exactly one point

ÞThe linear polynomial ax + b, a ≠ 0, has exactly one zero,
namely, the x-coordinate of the point (-b/a , 0) where the graph of y = ax + b
intersects the x-axis.

Þ The graph of equation y = ax+ + b
x + c has one of the two shapes either open upwards like È

(a >
0 ) or open downwards like Ç ( a < 0) . These curves are
called **parabolas**.

Þ The zeroes of a quadratic polynomial ax^{2} +
bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola
representing y = ax^{2} + bx + c intersects the x-axis.

Þ In general, given a polynomial p(x) of degree n, the graph
of y = p(x) intersects the x- axis at atmost n[n or
less than n] points.

Þ A polynomial p(x) of degree n has at most n zeroes.

Þif α and β are the zeroes of the quadratic
polynomial p(x) = ax^{2} + b
x + c, a ≠ 0, then you know that x – α
and x – β are the factors of p(x). Therefore,

ax^{2} + bx + c = a(x – α)
(x – β)= ax^{2} – a(α + β)x + a α β]

Comparing the coefficients of x^{2}, x and
constant terms on both the sides, we get , a = k**, **b =
– k(α + β) and c = kαβ.

This
gives

** **α + β = -b/a ; α β = c/a

Þ Relationship between the zeroes of a cubic polynomial and
its coefficients of

ax^{3} + bx^{2} + c x + d= a(x-a)(x-b)(x-g)

= ax^{3} – a(a+b+g)x^{2 }+ a(ab +bg+ga) - aabg

Comparing the coefficients of terms on both the sides, we
get, α + β + γ = –b/a; α
β + β γ + γ α =c/a; α β γ =– d/a

Þ Division algorithm states that given any polynomial *p*(*x*)
and any non-zero polynomial *g*(*x*), there are
polynomials *q*(*x*) and *r*(*x*) such that *p*(*x*)
= *g*(*x*) *q*(*x*) + *r*(*x*),
where *r*(*x*) = 0 or degree *r*(*x*) <
degree *g*(*x*).

Check point [Formative Assessment]

Q. 1. Find the zeroes of the quadratic polynomial x^{2} +
7x + 10, and verify the relationship between the zeroes and the coefficients.

Q.2. Find the zeroes of the polynomial x^{2} – 3 and
verify the relationship between the zeroes and the coefficients.

Q.3. Find a quadratic polynomial, the sum and product of whose
zeroes are – 3 and 2, respectively.

Q.4. Verify that 3, –1, -1/3 are the zeroes of the
cubic polynomial p(x) = 3x^{3 }– 5x^{2} –
11x – 3, and then verify the relationship between the zeroes and the
coefficients.

Q.5. Find a quadratic polynomial if the sum and product of its
zeroes respectively √2, 1/3

Q.6. Find all the zeroes of 2x^{4 }– 3x^{3} –
3x^{2} + 6x – 2, if you know that two of its zeroes are √2 and − √2

Q.7. Obtain all other zeroes of 3x^{4} + 6x^{3} –
2x^{2} – 10x – 5, if two of its zeroes are √(5/3) ,
-√(5/3)

Q.8. Find a cubic polynomial with the sum, sum of the product of
its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14
respectively

Q.9. If the zeroes of the polynomial *x*^{3} –
3*x*^{2} + *x *+ 1 are *a *– *b*, *a*, *a *+ *b*,
find *a *and *b*.

Q.10.** **If two zeroes of the polynomial *x*^{4} –
6*x*^{3} – 26*x*^{2} + 138*x *–
35 are 2 ± √3 , find other zeroes.

Q.11.** **If the polynomial *x*^{4} –
6*x*^{3} + 16*x*^{2} – 25*x *+ 10
is divided by another polynomial *x*^{2} – 2*x *+ *k*,
the remainder comes out to be *x *+ *a*, find *k *and *a*.

Q.12. If α and β are the zeros of the quadratic polynomial *f(x)
= x*^{2} - p (x+1) - c, Show that (α+ 1) (β + 1) = 1- c.

Class X Polynomial Test Paper-1

Class X Polynomial Test Paper-2

Class X Polynomial Test Paper-3

Class X Polynomial Test Paper-4

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