Answer :
To solve the problem of dividing the polynomial [tex]\( f(x) = 8x^5 + 43x^4 - 58x^3 + 60x^2 - 70 \)[/tex] by [tex]\( x - 4 \)[/tex], we will perform polynomial long division. Here's a step-by-step explanation:
1. Setup the Division:
- Write the dividend [tex]\( f(x) = 8x^5 + 43x^4 - 58x^3 + 60x^2 + 0x - 70 \)[/tex].
- The divisor is [tex]\( x - 4 \)[/tex].
2. Perform Division:
- Step 1: Divide the first term of the dividend [tex]\( 8x^5 \)[/tex] by the first term of the divisor [tex]\( x \)[/tex], which gives [tex]\( 8x^4 \)[/tex].
- Multiply [tex]\( 8x^4 \)[/tex] by the entire divisor [tex]\( x - 4 \)[/tex], resulting in [tex]\( 8x^5 - 32x^4 \)[/tex].
- Subtract this from the original polynomial:
[tex]\[
(8x^5 + 43x^4) - (8x^5 - 32x^4) = 75x^4
\][/tex]
- Bring down the next term [tex]\(-58x^3\)[/tex] to get [tex]\( 75x^4 - 58x^3 \)[/tex].
- Step 2: Divide [tex]\( 75x^4 \)[/tex] by [tex]\( x \)[/tex], yielding [tex]\( 75x^3 \)[/tex].
- Multiply [tex]\( 75x^3 \)[/tex] by [tex]\( x - 4 \)[/tex], giving [tex]\( 75x^4 - 300x^3 \)[/tex].
- Subtract:
[tex]\[
(75x^4 - 58x^3) - (75x^4 - 300x^3) = 242x^3
\][/tex]
- Bring down [tex]\( 60x^2 \)[/tex] to get [tex]\( 242x^3 + 60x^2 \)[/tex].
- Step 3: Divide [tex]\( 242x^3 \)[/tex] by [tex]\( x \)[/tex], getting [tex]\( 242x^2 \)[/tex].
- Multiply [tex]\( 242x^2 \)[/tex] by [tex]\( x - 4 \)[/tex], producing [tex]\( 242x^3 - 968x^2 \)[/tex].
- Subtract:
[tex]\[
(242x^3 + 60x^2) - (242x^3 - 968x^2) = 1028x^2
\][/tex]
- Bring down [tex]\( 0x \)[/tex] to get [tex]\( 1028x^2 + 0x \)[/tex].
- Step 4: Divide [tex]\( 1028x^2 \)[/tex] by [tex]\( x \)[/tex], yielding [tex]\( 1028x \)[/tex].
- Multiply [tex]\( 1028x \)[/tex] by [tex]\( x - 4 \)[/tex], giving [tex]\( 1028x^2 - 4112x \)[/tex].
- Subtract:
[tex]\[
(1028x^2 + 0x) - (1028x^2 - 4112x) = 4112x
\][/tex]
- Bring down [tex]\(-70\)[/tex] to get [tex]\( 4112x - 70 \)[/tex].
- Step 5: Divide [tex]\( 4112x \)[/tex] by [tex]\( x \)[/tex], resulting in [tex]\( 4112 \)[/tex].
- Multiply [tex]\( 4112 \)[/tex] by [tex]\( x - 4 \)[/tex], getting [tex]\( 4112x - 16448 \)[/tex].
- Subtract:
[tex]\[
(4112x - 70) - (4112x - 16448) = 16378
\][/tex]
3. Conclusion:
- The quotient is [tex]\( 8x^4 + 75x^3 + 242x^2 + 1028x + 4112 \)[/tex].
- The remainder is [tex]\( 16378 \)[/tex].
So, [tex]\( f(x) \)[/tex] divided by [tex]\( x - 4 \)[/tex] gives a quotient of [tex]\( 8x^4 + 75x^3 + 242x^2 + 1028x + 4112 \)[/tex] with a remainder of [tex]\( 16378 \)[/tex].
1. Setup the Division:
- Write the dividend [tex]\( f(x) = 8x^5 + 43x^4 - 58x^3 + 60x^2 + 0x - 70 \)[/tex].
- The divisor is [tex]\( x - 4 \)[/tex].
2. Perform Division:
- Step 1: Divide the first term of the dividend [tex]\( 8x^5 \)[/tex] by the first term of the divisor [tex]\( x \)[/tex], which gives [tex]\( 8x^4 \)[/tex].
- Multiply [tex]\( 8x^4 \)[/tex] by the entire divisor [tex]\( x - 4 \)[/tex], resulting in [tex]\( 8x^5 - 32x^4 \)[/tex].
- Subtract this from the original polynomial:
[tex]\[
(8x^5 + 43x^4) - (8x^5 - 32x^4) = 75x^4
\][/tex]
- Bring down the next term [tex]\(-58x^3\)[/tex] to get [tex]\( 75x^4 - 58x^3 \)[/tex].
- Step 2: Divide [tex]\( 75x^4 \)[/tex] by [tex]\( x \)[/tex], yielding [tex]\( 75x^3 \)[/tex].
- Multiply [tex]\( 75x^3 \)[/tex] by [tex]\( x - 4 \)[/tex], giving [tex]\( 75x^4 - 300x^3 \)[/tex].
- Subtract:
[tex]\[
(75x^4 - 58x^3) - (75x^4 - 300x^3) = 242x^3
\][/tex]
- Bring down [tex]\( 60x^2 \)[/tex] to get [tex]\( 242x^3 + 60x^2 \)[/tex].
- Step 3: Divide [tex]\( 242x^3 \)[/tex] by [tex]\( x \)[/tex], getting [tex]\( 242x^2 \)[/tex].
- Multiply [tex]\( 242x^2 \)[/tex] by [tex]\( x - 4 \)[/tex], producing [tex]\( 242x^3 - 968x^2 \)[/tex].
- Subtract:
[tex]\[
(242x^3 + 60x^2) - (242x^3 - 968x^2) = 1028x^2
\][/tex]
- Bring down [tex]\( 0x \)[/tex] to get [tex]\( 1028x^2 + 0x \)[/tex].
- Step 4: Divide [tex]\( 1028x^2 \)[/tex] by [tex]\( x \)[/tex], yielding [tex]\( 1028x \)[/tex].
- Multiply [tex]\( 1028x \)[/tex] by [tex]\( x - 4 \)[/tex], giving [tex]\( 1028x^2 - 4112x \)[/tex].
- Subtract:
[tex]\[
(1028x^2 + 0x) - (1028x^2 - 4112x) = 4112x
\][/tex]
- Bring down [tex]\(-70\)[/tex] to get [tex]\( 4112x - 70 \)[/tex].
- Step 5: Divide [tex]\( 4112x \)[/tex] by [tex]\( x \)[/tex], resulting in [tex]\( 4112 \)[/tex].
- Multiply [tex]\( 4112 \)[/tex] by [tex]\( x - 4 \)[/tex], getting [tex]\( 4112x - 16448 \)[/tex].
- Subtract:
[tex]\[
(4112x - 70) - (4112x - 16448) = 16378
\][/tex]
3. Conclusion:
- The quotient is [tex]\( 8x^4 + 75x^3 + 242x^2 + 1028x + 4112 \)[/tex].
- The remainder is [tex]\( 16378 \)[/tex].
So, [tex]\( f(x) \)[/tex] divided by [tex]\( x - 4 \)[/tex] gives a quotient of [tex]\( 8x^4 + 75x^3 + 242x^2 + 1028x + 4112 \)[/tex] with a remainder of [tex]\( 16378 \)[/tex].
