Answer :
To find the quotient of the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] divided by [tex]\( x^3 - 3 \)[/tex], we can perform polynomial long division. Here’s a step-by-step guide to solving this:
1. Setup for Division:
- We are dividing the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] (dividend) by [tex]\( x^3 - 3 \)[/tex] (divisor).
2. Divide the Leading Terms:
- Look at the leading term of the dividend, which is [tex]\( x^4 \)[/tex], and the leading term of the divisor, which is [tex]\( x^3 \)[/tex].
- Divide [tex]\( x^4 \)[/tex] by [tex]\( x^3 \)[/tex] to get [tex]\( x \)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x^4 - 3x \)[/tex].
- Subtract [tex]\( x^4 - 3x \)[/tex] from the original dividend [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex].
4. Calculate the New Dividend:
- After subtraction, you get a new polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + -3x + 3x - 15
\][/tex]
- Simplify to get: [tex]\( 5x^3 - 15 \)[/tex].
5. Repeat the Process:
- Now, divide the leading term of the new dividend [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex] to get [tex]\( 5 \)[/tex].
- Multiply the divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( 5 \)[/tex], resulting in [tex]\( 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from the new dividend [tex]\( 5x^3 - 15 \)[/tex], which gives a remainder of zero.
6. Obtain the Quotient:
- The combination of terms from the division steps gives the quotient. Thus, the quotient is [tex]\( x + 5 \)[/tex].
There you go, the quotient is [tex]\( x + 5 \)[/tex], and the remainder is zero, indicating a perfect division.
1. Setup for Division:
- We are dividing the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] (dividend) by [tex]\( x^3 - 3 \)[/tex] (divisor).
2. Divide the Leading Terms:
- Look at the leading term of the dividend, which is [tex]\( x^4 \)[/tex], and the leading term of the divisor, which is [tex]\( x^3 \)[/tex].
- Divide [tex]\( x^4 \)[/tex] by [tex]\( x^3 \)[/tex] to get [tex]\( x \)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x^4 - 3x \)[/tex].
- Subtract [tex]\( x^4 - 3x \)[/tex] from the original dividend [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex].
4. Calculate the New Dividend:
- After subtraction, you get a new polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + -3x + 3x - 15
\][/tex]
- Simplify to get: [tex]\( 5x^3 - 15 \)[/tex].
5. Repeat the Process:
- Now, divide the leading term of the new dividend [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex] to get [tex]\( 5 \)[/tex].
- Multiply the divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( 5 \)[/tex], resulting in [tex]\( 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from the new dividend [tex]\( 5x^3 - 15 \)[/tex], which gives a remainder of zero.
6. Obtain the Quotient:
- The combination of terms from the division steps gives the quotient. Thus, the quotient is [tex]\( x + 5 \)[/tex].
There you go, the quotient is [tex]\( x + 5 \)[/tex], and the remainder is zero, indicating a perfect division.
