Question

When a polynomial 6x4+8x3+29x2+21x+7 is divide by another polynomial 3x2+4x+1 the remainder in the form of ax+b .find a and b

Answer 1

Values are $\bf \red{a = - 15} \: and \: \green{b = - 2}$.

Step-by-step explanation:

Given:

• A polynomial $6 {x}^{4} + 8 {x}^{3} + 29 {x}^{2} + 21x + 7$ and
• Other polynomial $3 {x}^{2} + 4x + 1$.
• On divide the polynomial the remainder is in the form ax+b.

To find:

• Find the value of a and b.

Solution:

• Divide the former polynomial by later one.
• Compare the actual remainder with ax+b.
• Find the value of a and b.

Step 1:

Let $f(x) = 6 {x}^{4} + 8 {x}^{3} + 29 {x}^{2} + 21x + 7$

and

$g(x) = 3 {x}^{2} + 4x + 1$

Divide f(x) by g(x).

$3 {x}^{2} + 4x + 1 \: ) \: 6 {x}^{4} + 8 {x}^{3} + 29 {x}^{2} + 21x + 7 \: ( 2 {x}^{2} + 9 \\ 6 {x}^{4} + 8 {x}^{3} + 2 {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \\ ( - ) \: \: ( - ) \: \: ( - ) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ - - - - - - - - - \: \: \: \: \\ 27 {x}^{2} + 21x + 7 \\ 27 {x}^{2} + 36x + 9 \\ ( - ) \: \: \: ( - ) \: \: \: ( - ) \: \: \: \\ - - - - - - - - \: \: \: \\ - 15x - 2 \\ - - - - - - - - \: \:$

Remainder is $\bf - 15x - 2$

Step 2:

Compare remainder of step 1 with ax+b.

Compare actual remainder with given remainder.

$- 15x - 2 = ax + b$

So,

$\bf a = - 15$

and

$\bf b = - 2$

Thus,

Values are $\bf \red{a = - 15} \: and \: \green{b = - 2}$

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