Question

Using long division method, check whether the second polynomial is the factor of first polynomial. Need it Urgently.

f(x) = 6x³ + 19x² + 13x + (-3)
d(x) = 2x + 3
​

Answer 1

AnswEr :

We Can Divide any Polynomial in this Form :

◗ f(x) ÷ d(x) = q(x) with a remainder of r(x)

• We Can Write this as :

$\sf\underbrace{Dividend}_{\large f(x)} = \underbrace{Divisor}_{\large d(x)}\times\underbrace{Quotient}_{\large q(x)} + \underbrace{Remainder}_{\large r(x)}$

$\rule{300}{1}$

• Let's Head to the Question Now :

◗ f(x) = 6x³ + 19x² + 13x + (- 3)

⠀ ⠀⠀ = 6x³ + 19x² + 13x - 3

◗ d(x) = 2x + 3

$\boxed{\begin{minipage}{7 cm}\quad\begin{array}{m{3.5em}cccc}&& 3x^2& +5x&-1\\\cline{1-6}\multicolumn{2}{l}{2x+3\big)}&6x^3&+19x^2&+13x&-3\\&& -(6x^3&+9x^2)&&\\\cline{3-4}&&&10x^2&+13x&\\&&&-(10x^2&+15x) &\\\cline{4-5}&&&& -2x&-3\\&&&&-(-2x&-3)\\\cline{5-6}&&&&&0\\\end{array}\end{minipage}}$

As, we can see that the r(x) i.e. Remainder is equal to Zero ( 0 ).

∴ We can say that (2x + 3) is the Factor of 6x³ + 19x² + 13x + (-3).

$\rule{300}{2}$

• LONG METHOD DIVISION :

$\star\:\underline\text{Steps of Division of a Polynomial :}$

⋆ Arrange the indices of the polynomial in descending order. Replace the missing term(s) with 0.

⋆ Divide the first term of the dividend (the polynomial to be divided) by the first term of the divisor. This gives the first term of the quotient.

⋆ Multiply the divisor by the first term of the quotient.

⋆ Subtract the product from the dividend then bring down the next term. The difference and the next term will be the new dividend. Note: Remember the rule in subtraction "change the sign of the subtrahend then proceed to addition".

⋆ Repeat step 2 – 4 to find the second term of the quotient.

⋆ Continue the process until a remainder is obtained. This can be zero or is of lower index than the divisor.

If the divisor is a factor of the dividend, you will obtain a remainder equal to zero. If the divisor is not a factor of the dividend, you will obtain a remainder whose index is lower than the index of the divisor.

#answerwithquality #BAL

Answer 2

Answer:

Refer to the Attachment for your Answer.

Mark as Brainlesst. Thanks