Answer :
Sure! Let's walk through the long division of polynomials process to divide [tex]\(6x^5 - 7x^4 - 9x^3 + 7x^2 - 2x + 1\)[/tex] by [tex]\(2x + 5\)[/tex].
Step 1: Setup the Division
Write down the dividend [tex]\(6x^5 - 7x^4 - 9x^3 + 7x^2 - 2x + 1\)[/tex] and the divisor [tex]\(2x + 5\)[/tex].
Step 2: Divide the First Term
Divide the first term of the dividend, [tex]\(6x^5\)[/tex], by the first term of the divisor, [tex]\(2x\)[/tex].
[tex]\[
\frac{6x^5}{2x} = 3x^4
\][/tex]
Step 3: Multiply and Subtract
Multiply the entire divisor [tex]\(2x + 5\)[/tex] by [tex]\(3x^4\)[/tex] and subtract the result from the dividend:
[tex]\[
(2x + 5) \times 3x^4 = 6x^5 + 15x^4
\][/tex]
Subtract from the original polynomial:
[tex]\[
(6x^5 - 7x^4) - (6x^5 + 15x^4) = -22x^4
\][/tex]
Bring down the next term [tex]\(-9x^3\)[/tex] from the dividend, leading to:
[tex]\[
-22x^4 - 9x^3
\][/tex]
Step 4: Repeat the Process
1. Divide [tex]\(-22x^4\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{-22x^4}{2x} = -11x^3
\][/tex]
2. Multiply the entire divisor [tex]\(2x + 5\)[/tex] by [tex]\(-11x^3\)[/tex], and subtract:
[tex]\[
(2x + 5) \times -11x^3 = -22x^4 - 55x^3
\][/tex]
Subtract from [tex]\(-22x^4 - 9x^3\)[/tex]:
[tex]\[
(-22x^4 - 9x^3) - (-22x^4 - 55x^3) = 46x^3
\][/tex]
Bring down [tex]\(+7x^2\)[/tex]:
[tex]\[
46x^3 + 7x^2
\][/tex]
Step 5: Repeat Again
1. Divide [tex]\(46x^3\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{46x^3}{2x} = 23x^2
\][/tex]
2. Multiply:
[tex]\[
(2x + 5) \times 23x^2 = 46x^3 + 115x^2
\][/tex]
Subtract:
[tex]\[
(46x^3 + 7x^2) - (46x^3 + 115x^2) = -108x^2
\][/tex]
Bring down [tex]\(-2x\)[/tex]:
[tex]\[
-108x^2 - 2x
\][/tex]
Step 6: Continue
1. Divide [tex]\(-108x^2\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{-108x^2}{2x} = -54x
\][/tex]
2. Multiply:
[tex]\[
(2x + 5) \times -54x = -108x^2 - 270x
\][/tex]
Subtract:
[tex]\[
(-108x^2 - 2x) - (-108x^2 - 270x) = 268x
\][/tex]
Bring down [tex]\(+1\)[/tex]:
[tex]\[
268x + 1
\][/tex]
Step 7: Final Step
1. Divide [tex]\(268x\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{268x}{2x} = 134
\][/tex]
2. Multiply:
[tex]\[
(2x + 5) \times 134 = 268x + 670
\][/tex]
Subtract:
[tex]\[
(268x + 1) - (268x + 670) = -669
\][/tex]
This gives us the complete result:
Quotient: [tex]\(3x^4 - 11x^3 + 23x^2 - 54x + 134\)[/tex]
Remainder: [tex]\(-669\)[/tex]
Thus, the division can be expressed as:
[tex]\[
6x^5 - 7x^4 - 9x^3 + 7x^2 - 2x + 1 = (2x + 5)(3x^4 - 11x^3 + 23x^2 - 54x + 134) - 669
\][/tex]
Our remainder is [tex]\(-669\)[/tex], so we express it as [tex]\(-\frac{669}{2x + 5}\)[/tex].
For part (a) of your question, based on the remainder being negative [tex]\(-\frac{669}{2x + 5}\)[/tex], we would color the background yellow.
Step 1: Setup the Division
Write down the dividend [tex]\(6x^5 - 7x^4 - 9x^3 + 7x^2 - 2x + 1\)[/tex] and the divisor [tex]\(2x + 5\)[/tex].
Step 2: Divide the First Term
Divide the first term of the dividend, [tex]\(6x^5\)[/tex], by the first term of the divisor, [tex]\(2x\)[/tex].
[tex]\[
\frac{6x^5}{2x} = 3x^4
\][/tex]
Step 3: Multiply and Subtract
Multiply the entire divisor [tex]\(2x + 5\)[/tex] by [tex]\(3x^4\)[/tex] and subtract the result from the dividend:
[tex]\[
(2x + 5) \times 3x^4 = 6x^5 + 15x^4
\][/tex]
Subtract from the original polynomial:
[tex]\[
(6x^5 - 7x^4) - (6x^5 + 15x^4) = -22x^4
\][/tex]
Bring down the next term [tex]\(-9x^3\)[/tex] from the dividend, leading to:
[tex]\[
-22x^4 - 9x^3
\][/tex]
Step 4: Repeat the Process
1. Divide [tex]\(-22x^4\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{-22x^4}{2x} = -11x^3
\][/tex]
2. Multiply the entire divisor [tex]\(2x + 5\)[/tex] by [tex]\(-11x^3\)[/tex], and subtract:
[tex]\[
(2x + 5) \times -11x^3 = -22x^4 - 55x^3
\][/tex]
Subtract from [tex]\(-22x^4 - 9x^3\)[/tex]:
[tex]\[
(-22x^4 - 9x^3) - (-22x^4 - 55x^3) = 46x^3
\][/tex]
Bring down [tex]\(+7x^2\)[/tex]:
[tex]\[
46x^3 + 7x^2
\][/tex]
Step 5: Repeat Again
1. Divide [tex]\(46x^3\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{46x^3}{2x} = 23x^2
\][/tex]
2. Multiply:
[tex]\[
(2x + 5) \times 23x^2 = 46x^3 + 115x^2
\][/tex]
Subtract:
[tex]\[
(46x^3 + 7x^2) - (46x^3 + 115x^2) = -108x^2
\][/tex]
Bring down [tex]\(-2x\)[/tex]:
[tex]\[
-108x^2 - 2x
\][/tex]
Step 6: Continue
1. Divide [tex]\(-108x^2\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{-108x^2}{2x} = -54x
\][/tex]
2. Multiply:
[tex]\[
(2x + 5) \times -54x = -108x^2 - 270x
\][/tex]
Subtract:
[tex]\[
(-108x^2 - 2x) - (-108x^2 - 270x) = 268x
\][/tex]
Bring down [tex]\(+1\)[/tex]:
[tex]\[
268x + 1
\][/tex]
Step 7: Final Step
1. Divide [tex]\(268x\)[/tex] by [tex]\(2x\)[/tex]:
[tex]\[
\frac{268x}{2x} = 134
\][/tex]
2. Multiply:
[tex]\[
(2x + 5) \times 134 = 268x + 670
\][/tex]
Subtract:
[tex]\[
(268x + 1) - (268x + 670) = -669
\][/tex]
This gives us the complete result:
Quotient: [tex]\(3x^4 - 11x^3 + 23x^2 - 54x + 134\)[/tex]
Remainder: [tex]\(-669\)[/tex]
Thus, the division can be expressed as:
[tex]\[
6x^5 - 7x^4 - 9x^3 + 7x^2 - 2x + 1 = (2x + 5)(3x^4 - 11x^3 + 23x^2 - 54x + 134) - 669
\][/tex]
Our remainder is [tex]\(-669\)[/tex], so we express it as [tex]\(-\frac{669}{2x + 5}\)[/tex].
For part (a) of your question, based on the remainder being negative [tex]\(-\frac{669}{2x + 5}\)[/tex], we would color the background yellow.
