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Answer :
To find the quotient when [tex]\( 4x^2 - 36 \)[/tex] is divided by [tex]\( x + 3 \)[/tex], follow these steps:
1. Identify the terms in the polynomial: We have the dividend [tex]\( 4x^2 - 36 \)[/tex] and the divisor [tex]\( x + 3 \)[/tex].
2. Perform polynomial long division:
- Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{4x^2}{x} = 4x
\][/tex]
So, the first term of the quotient is [tex]\( 4x \)[/tex].
- Multiply the entire divisor [tex]\( x + 3 \)[/tex] by [tex]\( 4x \)[/tex]:
[tex]\[
(4x)(x + 3) = 4x^2 + 12x
\][/tex]
- Subtract this result from the original dividend:
[tex]\[
(4x^2 - 36) - (4x^2 + 12x) = -12x - 36
\][/tex]
3. Continue the division process:
- Divide the first term of the new polynomial by the first term of the divisor:
[tex]\[
\frac{-12x}{x} = -12
\][/tex]
So, the next term of the quotient is [tex]\(-12\)[/tex].
- Multiply the entire divisor [tex]\( x + 3 \)[/tex] by [tex]\(-12\)[/tex]:
[tex]\[
(-12)(x + 3) = -12x - 36
\][/tex]
- Subtract this result from the current polynomial:
[tex]\[
(-12x - 36) - (-12x - 36) = 0
\][/tex]
4. Obtain the final results:
- The quotient is the sum of the terms we found:
[tex]\[
4x - 12
\][/tex]
- The remainder is [tex]\( 0 \)[/tex].
Therefore, the quotient when [tex]\( 4x^2 - 36 \)[/tex] is divided by [tex]\( x + 3 \)[/tex] is [tex]\( 4x - 12 \)[/tex].
1. Identify the terms in the polynomial: We have the dividend [tex]\( 4x^2 - 36 \)[/tex] and the divisor [tex]\( x + 3 \)[/tex].
2. Perform polynomial long division:
- Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{4x^2}{x} = 4x
\][/tex]
So, the first term of the quotient is [tex]\( 4x \)[/tex].
- Multiply the entire divisor [tex]\( x + 3 \)[/tex] by [tex]\( 4x \)[/tex]:
[tex]\[
(4x)(x + 3) = 4x^2 + 12x
\][/tex]
- Subtract this result from the original dividend:
[tex]\[
(4x^2 - 36) - (4x^2 + 12x) = -12x - 36
\][/tex]
3. Continue the division process:
- Divide the first term of the new polynomial by the first term of the divisor:
[tex]\[
\frac{-12x}{x} = -12
\][/tex]
So, the next term of the quotient is [tex]\(-12\)[/tex].
- Multiply the entire divisor [tex]\( x + 3 \)[/tex] by [tex]\(-12\)[/tex]:
[tex]\[
(-12)(x + 3) = -12x - 36
\][/tex]
- Subtract this result from the current polynomial:
[tex]\[
(-12x - 36) - (-12x - 36) = 0
\][/tex]
4. Obtain the final results:
- The quotient is the sum of the terms we found:
[tex]\[
4x - 12
\][/tex]
- The remainder is [tex]\( 0 \)[/tex].
Therefore, the quotient when [tex]\( 4x^2 - 36 \)[/tex] is divided by [tex]\( x + 3 \)[/tex] is [tex]\( 4x - 12 \)[/tex].
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