Question

Divide the polynomial P (X) by the polynomial G (X) and find the question and remainder in each of the following :-

1. P(x) = X³-3x²+5x-3 , G(x) = x²-2.
2. P(x) = X⁴-3x²+4x+5 , G(x) = x²+1-x.
3. P(x) = X⁴-5x+6 , G( x) = 2-x².

How to solve this Question. ​

Answer 1

$\large\underline{\text{Polynomial long division}}$

We solve the question through the method of long division. When we divide a number by another number, we try to eliminate the largest number. This is common in polynomial division, as the target is to eliminate the highest degree term.

①$P(x)\text{ by }G(x)$

$\iff x^{3}-3x^{2}+5x-3=(x^{2}-2)(x-3)+7x+9$

②$P(x)\text{ by }G(x)$

$\iff x^{4}-3x^{2}+4x+5=(x^{2}-x+1)(x^{2}+x-3)+8$

③$P(x)\text{ by }G(x)$

$\iff x^{4}-5x+6=(-x^{2}+2)(-x^{2}-2)-5x+10$

If we list them as quotients and remainders:-

①Quotient is $x-3$. Remainder is $7x+9$.

②Quotient is $x^{2}+x-3$. Remainder is $8$.

③Quotient is $-x^{2}-2$. Remainder is $-5x+10$.

Here, I included the process of your each polynomial division. The method is  using only coefficients in polynomial division. It is simpler and takes less time. :)

$\large\underline{\text{More information}}$

Suppose we have divided the polynomial. If $A$ and $B$ are the dividend and divisor, $Q$ and $R$ are the quotient and remainder, we would get the following equation:-

$\large\boxed{A=BQ+R}$

Such type of equation is called the identity. The left and right hand side are always equal. This fact is used in dealing with polynomials.

Answer 2

Answer:

${\large{\bold{\purple{\underline{Solution}}}}}$

Question :-

Divide the polynomial P (X) by the polynomial G (X) and find the question and remainder in each of the following :-

• ☯ P(x) = X³-3x²+5x-3 , G(x) = x²-2.
• ☯ P(x) = X⁴-3x²+4x+5 , G(x) = x²+1-x.
• ☯ P(x) = X⁴-5x+6 , G( x) = 2-x².

Solution :-

1. P(x) = X³-3x²+5x-3 , G(x) = x²-2.

➙ Quotient is x - 3

➙ Remainder is 7x + 9

2. P(x) = X⁴-3x²+4x+5 , G(x) = x²+1-x.

➙ Quotient is x² + x - 3

➙ Remainder is 8

3. P(x) = X⁴-5x+6 , G( x) = 2-x².

➙ Quotient is - x² - 2

➙ Remainder is - 5x + 10

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$\qquad\qquad\underline{\textsf{\textbf{ \color{magenta}{Brainly\: Extra\: Shots :-} }}}$

Polynomial :

• A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division.

Types of Polynomials :

❶ Monomial :-

• A monomial is an expression which contains only one term.

❷ Binomial :-

• A binomial is a polynomial expression which contains exactly two terms.

❸ Trinomial :-

• A trinomial is an expression which is composed of exactly three terms.