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Quadratic Equations

Quadratic Equations

MODULES

Basics of Polynomial & Quadratic Equations
Discriminant & Graphical Representation
Sum & Product of Roots
Factorisation Method
Formulation & Completion of Squares
Changes to Roots
Mistakes in Roots, Common Roots & Squaring
Infinite Series & Transposed
Other Types
Higher Order Equations
Synthetic Division & Remainder Theorem
Maxima, Minima & Descrates Rule
Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Quadratic Equations 1
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Quadratic Equations 2
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Quadratic Equations 3
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PRACTICE

Quadratic Equations : Level 1
Quadratic Equations : level 2
Quadratic Equations : level 3
ALL MODULES

CAT 2025 Lesson : Quadratic Equations - Synthetic Division & Remainder Theorem

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5.4 Division of Polynomials

Let's first look at normal division. Note that this can be used for dividing any
222 polynomials. Let's take a simple example of 5x3+6x2+3x−95x^{3} + 6x^{2} + 3x - 95x3+6x2+3x−9 divided by (x+2)(x + 2)(x+2). It involves the following steps.

1) First term in the quotient will be the term which when multiplied with the highest-degree term in the divisor (i.e. xxx) will produce the highest-degree term in the dividend (5x3)(5x^{3})(5x3).

2) Write down the remainder after subtracting the product of the first term and the divisor from the dividend.

3) Second term in the quotient will be the term which when multiplied with the highest-degree term in the divisor (i.e. xxx) will produce the highest-degree term in the dividend and this process continues.

4) This process continues till the degree of the remainder is lower than that of the divisor.

(When the divisor has a degree of
111, then the remainder will have a degree of 111 less than the divisor, which is 000. Therefore, the remainder will be a constant.)

In the case to the right, when
5x3+6x2+3x−95x^{3} + 6x^{2} + 3x - 95x3+6x2+3x−9 is divided by (x+2)(x + 2)(x+2), we get a quotient of 5x2−4x+115x^{2} - 4x + 115x2−4x+11 and a remainder of −31-31−31.



5.5 Remainder Theorem

For any real number
aaa, the remainder when a polynomial f(x)f(x)f(x) is divided by (x−a)(x - a)(x−a) is f(a)f(a)f(a).

Taking the above example, when f(x)
=== 5x3+6x2+3x−95x^{3} + 6x^{2} + 3x - 95x3+6x2+3x−9 is divided by (x+2)(x + 2)(x+2),

Remainder
=f(−2)=5(−2)3+6(−2)2+3(−2)−9= f(-2) = 5(-2)^{3} + 6(-2)^{2} + 3(-2) - 9=f(−2)=5(−2)3+6(−2)2+3(−2)−9 =−40+24−6−9=−31= -40 + 24 - 6 - 9 = -31=−40+24−6−9=−31

This is definitely a quick way to find the remainder. However, we will not be able to find the quotient using this approach. The following process of synthetic division will help save time in these questions.

5.6 Synthetic Division

Synthetic division can be applied only when the divisor is of the form
(x−a)(x - a)(x−a). Using the same example of 5x3+6x2+3x−95x^{3} + 6x^{2} + 3x - 95x3+6x2+3x−9 divided by (x+2)(x + 2)(x+2), note the following steps.

1) aaa will be the divisor. Write the down the coefficients of each of the terms in decreasing order of their powers.

2) Fill
000 under the first coefficient and add the two values and write it below.

3) Multiply the first sum (555) with the divisor (−2-2−2) and fill the product of (−10-10−10) under the next coefficient (666). Once again, add the two values (-4) and write it below.

4) Repeat the process outlined in step 333 till the end. The right-most number (−31-31−31) in the bottom is the remainder. The other numbers to the left of it are the coefficients of the quotient whose degree will be 111 less than the dividend.

Note the the remainder in this case is
−31-31−31. The other numbers in the bottom (i.e., 5,−45, -45,−4 and 111111) form the coefficients of the quotient, i.e. 5x2−4x+115x^{2} - 4x + 115x2−4x+11.



Example 24

If 3x3−9x2+kx−123x^{3} - 9x^{2} + kx - 123x3−9x2+kx−12 is divisible by (x−3)(x - 3)(x−3), then it is also divisible by: [FMS 2010]

(1)
3x2−43x^{2} - 43x2−4            (2) 3x2+43x^{2} + 43x2+4            (3) 3x−43x - 43x−4            (4) 3x+43x + 43x+4           

Solution

Let f(x)=3x3−9x2+kx−12f(x) = 3x^{3} - 9x^{2} + kx -12f(x)=3x3−9x2+kx−12

As
(x−3)(x - 3)(x−3) is a factor, f(3)=3×33−9×32+3k−12f(3) = 3 \times 3^{3} - 9 \times 3^{2} + 3k - 12f(3)=3×33−9×32+3k−12
⇒
3k=123k = 123k=12
⇒
k=4k = 4k=4

∴
f(x)=3x3−9x2+4x−12f(x) = 3x^{3} - 9x^{2} + 4x - 12f(x)=3x3−9x2+4x−12

To find other factors, let's apply synthetic division to divide
f(x)f(x)f(x) by (x−3)(x - 3)(x−3) and find the quotient.



The coefficients of the quotient are
3,03, 03,0 and 444. Therefore, the quotient is 3x2+0×x+4=3x2+43x^{2} + 0 \times x + 4 = 3x^{2} + 43x2+0×x+4=3x2+4

When the remainder is 0, a dividend will also be divisible by its quotient. So,
f(x)f(x)f(x) is divisible by 3x2+43x^{2} + 43x2+4.

Answer: (2)
3x2+43x^{2} + 43x2+4

5.7 Factorising polynomials

A factor is a number that leaves a remainder of
000. So, to factorise a higher-order polynomial please apply the following

1) Questions with higher order polynomial will typically have at least
111 small integral root. So, start by applying remainder theorem and substitute values such as f(111), f(−1-1−1), f(222), f(−2-2−2), ..etc. and find the root where the remainder is 000

2) Once you've found a root, apply synthetic division and find the quotient.

3) Once again, apply remainder theorem on the quotient to find another root and then apply synthetic division to get the next quotient.

4) Continue this process till you're left with a quadratic equation (factorising of which has been explained earlier in this lesson).

Example 25

What is the sum of the squares of the roots of the equation 4x3+9x2−x−64x^{3} + 9x^{2} - x - 64x3+9x2−x−6 ?

Solution

Let f(x)=4x3+9x2−x−6f(x) = 4x^{3} + 9x^{2} - x - 6f(x)=4x3+9x2−x−6. We begin with checking if 1,−1,2,−2,1, -1, 2, -2,1,−1,2,−2, etc. are roots.

f(1)=6f(1) = 6f(1)=6
f(−1)=−4+9+1−6f(-1) = -4 + 9 + 1 - 6f(−1)=−4+9+1−6

∴
x=−1x = -1x=−1 is a root. We not apply synthetic division to find the quotient.



As the coefficients of the quotient are
4,54, 54,5 and −6-6−6,

4x3+9x2−x−6=(x+1)(4x2+5x−6)4x^{3} + 9x^{2} - x - 6 = (x + 1)(4x^{2} + 5x - 6)4x3+9x2−x−6=(x+1)(4x2+5x−6)
=(x+1)(4x2+8x−3x−6)= (x + 1)(4x^{2} + 8x - 3x - 6)=(x+1)(4x2+8x−3x−6) =(x+1)(4x(x+2)−3(x+2))= (x + 1)(4x(x + 2) - 3(x + 2))=(x+1)(4x(x+2)−3(x+2))
=(x+1)(4x−3)(x+2)= (x + 1)(4x - 3)(x + 2)=(x+1)(4x−3)(x+2)

Roots
=−1,34,−2= -1, \dfrac{3}{4}, -2=−1,43​,−2

Sum of Squares of roots
=1+916+4=8916= 1 + \dfrac{9}{16} + 4 = \dfrac{89}{16}=1+169​+4=1689​

Answer:
8916\dfrac{89}{16}1689​

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