# Chapter 2.Polynomials

**1.Important Identities of Polynomials**

1. (*x *+ *y*) ² = *x*² + 2*xy *+ *y*²

2 .(*x *– *y*) ² = *x*² – 2*xy *+ *y*²

3. *x*² – *y*² = (*x *+ *y*) (*x *– *y*)

**2.Use of Polynomial Identities**

1) factorisation.

2.find out Square of Big Numbers

*3.Remainder Theorem*

*4.* *Factor Theorem*

*Example:- ax+b+c=0 is monomial , degree of x is 1*

* ax*²+bx+c=0 is Binomial (or Quadratic Polynomial) , degree of x is 2

*ax*^{3}+*ax*²+bx+c=0 is cubic Polynomial, degree of x is 3.

**3.Remainder Theorem : ***Let p*(*x*) *be any polynomial of degree greater than or*

*equal to one and let a be any real number. If p*(*x*) *is divided by the linear*

*polynomial x – a, then the remainder is p*(*a*)*.*

**Proof : **Let *p*(*x*) be any polynomial with degree greater than or equal to 1. Suppose

that when *p*(*x*) is divided by *x *– *a*, the quotient is *q*(*x*) and the remainder is *r*(*x*), i.e.,

*p*(*x*) = (*x *– *a*) *q*(*x*) + *r*(*x*)

Since the degree of *x *– *a *is 1 and the degree of *r*(*x*) is less than the degree of *x *– *a*,

the degree of *r*(*x*) = 0. This means that *r*(*x*) is a constant, say *r*.

So, for every value of *x*, *r*(*x*) = *r*.

Therefore, *p*(*x*) = (*x *– *a*) *q*(*x*) + *r*

In particular, if *x *= *a*, this equation gives us

*p*(*a*) = (*a *– *a*) *q*(*a*) + *r*

= *r,*

which proves the theorem.

4. **Factor Theorem : **If *p*(*x*) is a polynomial of degree *n *> 1 and *a *is any real number,

then (i) *x *– *a *is a factor of *p*(*x*), if *p*(*a*) = 0, and (ii) *p*(*a*) = 0, if *x *– *a *is a factor of *p*(*x*).

Proof: By the Remainder Theorem, *p*(*x*)=(*x *– *a*) *q*(*x*) + *p*(*a*).

(i) If *p*(*a*) = 0, then *p*(*x*) = (*x *– *a*) *q*(*x*), which shows that *x *– *a *is a factor of *p*(*x*).

(ii) Since *x – a *is a factor of *p*(*x*), *p*(*x*) = (*x *– *a*) *g*(*x*) for same polynomial *g*(*x*).

In this case, *p*(*a*) = (*a *– *a*) *g*(*a*) = 0.

5. Factorisation of the polynomial *ax*2 + *bx *+ *c ***by splitting the middle term **is as

follows:

Let its factors be (*px *+ *q*) and (*rx *+ *s*). Then

*ax*2 + *bx *+ *c *= (*px *+ *q*) (*rx *+ *s*) = *pr x*2 + (*ps *+ *qr*) *x *+ *qs*

Comparing the coefficients of *x*2, we get *a *= *pr*.

Similarly, comparing the coefficients of *x*, we get *b *= *ps *+ *qr*.

And, on comparing the constant terms, we get *c *= *qs*.

This shows us that *b *is the sum of two numbers *ps *and *qr*, whose product is

(*ps*)(*qr*) = (*pr*)(*qs*) = *ac*.

Therefore, to factorise *ax*2 + *bx *+ *c*, we have to write *b *as the sum of two

numbers whose product is *ac*.

6.** Algebraic Identities**

* Identity I: *(a + b)

^{2}= a

^{2}+ 2ab + b

^{2}

*Identity II:** *(a – b)^{2} = a^{2} – 2ab + b^{2}

* Identity III: *a

^{2}– b

^{2}= (a + b)(a – b)

* Identity IV: *(x + a)(x + b) = x

^{2}+ (a + b) x + ab

* Identity V: *(a + b + c)

^{2}= a

^{2}+ b

^{2}+ c

^{2}+ 2ab + 2bc + 2ca

* Identity VI: *(a + b)

^{3}= a

^{3}+ b

^{3}+ 3ab (a + b)

*Identity VII:** *(a – b)^{3} = a^{3} – b^{3} – 3ab (a – b)

* Identity VIII: *a

^{3}+ b

^{3}+ c

^{3 }– 3abc = (a + b + c)(a

^{2}+ b

^{2}+ c

^{2}– ab – bc – ca)

Summary of this chapter

**1. **A *polynomial p*(*x*) in one variable *x *is an algebraic expression in *x *of the form

*p(x*) = *a**n**x**n *+ *a**n*–1*x**n *– 1 *+ . . . + a*2*x*2 + *a*1*x *+ *a*0,

where *a*0, *a*1, *a*2, . . ., *a**n *are constants and *a**n *¹ 0.

*a*0, *a*1, *a*2, . . ., *a**n *are respectively the *coefficients *of *x*0, *x*, *x*2, . . ., *x**n*, and *n *is called the *degree*

*of the polynomial*. Each of *a**n**x**n*, *a**n*–1 *x**n*–1, …, *a*0, with *a**n *¹ 0, is called a *term *of the polynomial

*p*(*x*).

**2. **A polynomial of one term is called a monomial.

**3. **A polynomial of two terms is called a binomial.

**4. **A polynomial of three terms is called a trinomial.

**5. **A polynomial of degree one is called a linear polynomial.

**6. **A polynomial of degree two is called a quadratic polynomial.

**7. **A polynomial of degree three is called a cubic polynomial.

**8. **A real number ‘*a*’ is a *zero *of a polynomial *p*(*x*) if *p*(*a*) = 0. In this case, *a *is also called a

*root *of the equation *p*(*x*) = 0.

**9. **Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial

has no zero, and every real number is a zero of the zero polynomial.

**10. **Remainder Theorem : If *p*(*x*) is any polynomial of degree greater than or equal to 1 and *p*(*x*)

is divided by the linear polynomial *x *– *a*, then the remainder is *p*(*a*).

**11. **Factor Theorem : *x *– *a *is a factor of the polynomial *p*(*x*), if *p*(*a*) = 0. Also, if *x *– *a *is a factor

of *p*(*x*), then *p*(*a*) = 0.

**12. **(*x *+ *y *+ *z*)2 = *x*2 + *y*2 + *z*2 + 2*xy *+ 2*yz *+ 2*zx*

**13. **(*x *+ *y*)3 = *x*3 + *y*3 + 3*xy*(*x *+ *y*)

**14. **(*x *– *y*)3 = *x*3 – *y*3 – 3*xy*(*x *– *y*)

**15. ***x*3 + *y*3 + *z*3 – 3*xyz *= (*x *+ *y *+ *z*) (*x*2 + *y*2 + *z*2 – *xy *– *yz *– *zx*)