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 Answer the following and justify:
CBSE | Class 10 | Exercise 2.2 | Maths | NCERT Exemplar | Polynomials
 Answer the following and justify:

(i) Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5? (ii) What will the quotient and remainder be on division of ax2 + bx + c by px3 + qx2 + rx + s, p ≠ 0?

 (I) Can x2 – 1 be the remainder on division of x6 + 2×3 + x – 1 by a polynomial in x of degree 5?

No, x2 – 1 can’t be the remainder on division of x6 + 2×3 + x – 1 by a polynomial in x of degree 5.

Avocation:

At the point when a degree 6 polynomial is isolated by degree 5 polynomial,

The remainder will be of degree 1.

Expect to be that (x2 – 1) partitions the degree 6 polynomial with and the remainder got is degree 5 polynomial (1)

As indicated by our supposition,

    \[\left( degree\text{ }6\text{ }polynomial \right)\text{ }=\text{ }\left( x2\text{ }\text{ }1 \right)\left( degree\text{ }5\text{ }polynomial \right)\text{ }+\text{ }r\left( x \right)\text{ }\left[ \text{ }Since,\text{ }\left( a\text{ }=\text{ }bq\text{ }+\text{ }r \right) \right]\]

    \[=\text{ }\left( degree\text{ }7\text{ }polynomial \right)\text{ }+\text{ }r\left( x \right)\text{ }\left[ \text{ }Since,\text{ }\left( x2\text{ }term\text{ }\times \text{ }x5\text{ }term\text{ }=\text{ }x7\text{ }term \right) \right]\]

    \[=\text{ }\left( degree\text{ }7\text{ }polynomial \right)\]

From the above condition, unmistakably, our supposition that is repudiated.

    \[x2\text{ }\text{ }1\]

can’t be the remainder on division of

    \[x6\text{ }+\text{ }2x3\text{ }+\text{ }x\text{ }\text{ }1\]

by a polynomial in x of degree 5

Consequently Proved.

(ii) What will the remainder and leftover portion be on division of

    \[ax2\text{ }+\text{ }bx\text{ }+\text{ }c\text{ }by\text{ }px3\text{ }+\text{ }qx2\text{ }+\text{ }rx\text{ }+\text{ }s,\text{ }p\text{ }\ne \text{ }0?\]

Arrangement:

Level of the polynomial

    \[px3\text{ }+\text{ }qx2\text{ }+\text{ }rx\text{ }+\text{ }s\text{ }is\text{ }3\]

Level of the polynomial

    \[ax2\text{ }+\text{ }bx\text{ }+\text{ }c\text{ }is\text{ }2\]

Here, level of px3 + qx2 + rx + s is more noteworthy than level of the

    \[ax2\text{ }+\text{ }bx\text{ }+\text{ }c\]

In this manner, the remainder would be zero, What’s more, the rest of be the

    \[profit\text{ }=\text{ }ax2\text{ }+\text{ }bx\text{ }+\text{ }c.\]

i

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