- Constants and Variable
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- Polynomial expression
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- how to find the degree of a polynomial
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- Value of the polynomial
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- Zeros or roots of the polynomial
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- Adding Polynomials
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- subtracing Polynomials
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- Multiplying Polynomials
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- Dividing Polynomails
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- How to factor polynomials
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- Solved Examples Polynomials
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- polynomial Formative Assignment
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- Dividing Polynomial worksheet
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- Factoring polynomial worksheet
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- Polynomial Test Paper

Multiplication of polynomial is simple task and we can follow below steps to multiply polynomials

- Arrange both the polynomial is same order of exponent . It would be good to have terms arrange from highest exponent to lowest exponent
- We have to distribute each term of the first polynomial to every term of the second polynomial.

a. when you multiply two terms together you must multiply the coefficient (numbers) and add the exponents

b. Also as we already know ++ equals =, +- or -+ equals - and -- equals + - group like terms

So multiplication may involve

- Multiplication of monomial to monomial
- Multiplication of monomial to binomial, trinomial or more terms polynomials
- Multiplication of binomial, trinomial or more terms polynomials to monomial
- Multiplication of binomial to binomial, trinomial or more terms polynomials
- Multiplication of trinomial to trinomial or more terms polynomials

- $x^2 \times (2x^{22}) \times (4x^26)$
- $2 (\frac {-10xy^3}{3}) \times (\frac {6x^3y}{5})$
- $( x) \times (x^2) \times (x^3) \times (x^8)$

We will use the below property extensively in above questions

$x^m \times x^n \times x^o=x^{m+n+o}$

1. As you know from above

So, we get

$x^2 \times (2x^{22}) \times (4x^{26}) =8a^{48}$

2. $2 (\frac {-10xy^3}{3}) \times (\frac {6x^3y}{5})$ $=(-8x^4y^4)$

3.$( x) \times (x^2) \times (x^3) \times (x^8)$

$=x^{14}$

(i) (2x + 5) and (4x - 3)

(ii) (x - 8) and (3x - 4)

(iii)(2.5x - 0.5) and (2.5x + 0.5)

Let $( a+b) (c+d)$ to be done

then

$( a+b) (c+d)= a(c+d) + b( c+d)$

$=(a \times c)+(a \times d)+(b \times c)+(b \times d)$

The above expression of multiplying binomials is called FOIL method

FOIL stands FIRST ($a \times c$) OUTER ($a \times d$) INNER ($b \times c $) LAST($b \times d$)

We will use the same concept in all the question below

1. $(2x + 5)(4x - 3)$

$=2x \times 4x - 2x \times 3 +5 \times 4x- 5 \times 3$

$=8x^2 - 6x + 20x -15$

$=8x^2 + 14x -15$

2. $( y - 8)(3y - 4)$

$= y \times 3y - 4y - 8 \times 3y + 32$

$= 3y^2- 4y - 24y + 32$

$= 3y^2- 28y + 32$

3. $(2.5l - 0.5m)(2.5l + 0.5)$

Using $(a+b)(a-b) = a^2- b^2$

We get ,

$= 6.25l^2- 0.25m^2$

$(3x + 2)(4x^2 - 7x + 5)$

Let $( a+b) (c+d+e)$ to be done

then

$( a+b) (c+d+e)= a(c+d+e) + b( c+d+e)$

$=(a \times c)+(a \times d)+ (a \times e)+ (b \times c)+(b \times d) +(b \times e) $

We will use the same concept in all the question below

$=3x(4x^2 - 7x + 5) + 2(4x^2 - 7x + 5)$

$=12x^3 -21x^2 +15x +8x^2 -14x+10$

$=12x^3 -13x^2 +x+10$

- $(x-5)(x-9)$
- $\left(x + 10\right) \left(x^{2} + x + 2\right)$
- $3x^2 \times 4x^{20}$
- $\left(x + 4\right) \left(x^{2} + 4 x + 7\right)$
- $(x+9)(x-6)$
- $\left(x + 2\right) \left(x + 4\right)^{2}$
- $\left(x + 8\right) \left(x^{2} + 2 x + 9\right)$
- $(x-2)(x-2)$

When a polynomial p(x) is divided by the polynomial g(x), we get quotient q(x) and remainder r(x)

$p(x)=g(x).q(x)+r(x)$

Notes

1. The degree of the reminder r(x) is always less then divisor g(x)

Now Let us see how to divide the polynomial by another non-zero polynomial

Steps to divide a polynomial by another polynomial. This is also called the long division method of polynomials

1. Arrange the term in decreasing order in both the polynomial

2. Divide the highest degree term of the dividend by the highest degree term of the divisor to obtain the first term

3. Now We multiply the divisor by the first term of the quotient, and subtract this product from the dividend

4. Similar steps are followed till we get the reminder whose degree is less than of divisor

Divide p(x) by g(x), where $p(x) = x + 4x^2 -1$ and $g(x) = 1 + x$

We carry out the process of division by means of the following steps:

Now the degree of the reminder is less than degree of the divisor so process stops here

Complete division is illustrated below

Quotient is 3x -3 and Reminder is 2

Now $4x^2 + x -1= (x+1)(4x-3) +2$

- Divide $x^{3} + 12 x^{2} + 39 x + 29$ by $(x+1)$
- Divide $x^{3} + 14 x^{2} + 43 x + 32$ by $(x+3)$
- Divide $x^{3} + 5 x^{2} + 6 x + 2$ by $(x+1)$
- Divide $x^{2} + 2 x - 48$ by $(x+8)$
- Divide $x^{3} + 12 x^{2} + 47 x + 60$ by $(x+3)$

- for (x-a) then remainder P(a)

- for (x+a) => x -(-a),then remainder will be P(-a)

- for (ax-b) => a(x- b/a) ,the remainder will be P(b/a)

- for (ax+b) => a(x+b/a),the remainder will be P(-b/a)

- for (b-ax)=> -a(x-b/a),the remainder will be P(b/a)

### Quiz Time

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Class 9 Maths Class 9 Science