If a, b, c, d are in GP then (a) a+b, b+c2 c+d are in GP (b) a x^2+c is a factor of a x^3+b x^2+

If a, b, c, d are in GP then (a) a+b, b+c2 c+d are in GP (b)...

Key Concepts

Algebraic Manipulation in Sequences

When dealing with sequences like a geometric progression, algebraic manipulation involves using the inherent relationships between the terms, often by substituting expressions involving the common ratio. This skill is crucial when testing if newly formed sequences or expressions (for instance, sums or polynomials constructed from sequence terms) retain certain properties, such as being in geometric progression or having specific factors.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials that, when multiplied together, yield the original polynomial. This concept is essential in solving polynomial equations and in analyzing the properties of algebraic expressions such as determining roots or potential simplifications.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. This leads to specific relationships among the terms that can be exploited to simplify or transform expressions involving these terms.

Factor Theorem

The factor theorem is a fundamental result in algebra which states that a polynomial f(x) has a factor (x - r) if and only if f(r) = 0. This theorem is key in determining the factors of a polynomial and is often used to verify divisibility and simplify polynomial expressions.

Transcript

00:01 I've given here ab cd is in gp right the first part says that a plus b plus b plus b plus b plus t plus gp we assume that it's in gp if we take here b plus t plus t f f f squared equals a plus b plus b c plus ab plus b c plus b c plus b c plus b c plus b b this we have now i can just work on here, ab cd is in gp, right? so we can take, let's say, a over b equals b over c, equals b over c, equals b over c, equals b over d, equals 1 over k.

00:51 So from here, we'll find out b equals ak, t equals ak square, and d equal akq.

01:02 Let's put the values here, ak in this, and c as ak square, and c, d as akq.

01:07 We'll find out that this left -hand side is equal to right -hand side.

01:11 It's both going to be equal.

01:13 That is only possible when we have a, b, c, d is in g -b.

01:16 If a -b -c -d is in g -b.

01:21 So, option a, here we have.

01:27 So option a is correct.

01:32 If we look at now, option 2, that is, from beginners, ax2 plus c is a factor of, given the qiquilomium, for b -part.

01:47 A x cubed plus b x squared plus d x plus b x squared plus d plus c plus c b over a so to check that a x square plus c is the factor of this this we can write it as a x square plus c times x plus b over a plus d negative c b over a now if we just work on the right side here right hand side will be a x cube plus we get a x square times b over it so bxp plus cx plus you will get c b over a and negative c b over a and plus b certifies you.

02:31 We can write in this way...

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