Answer :
Sure! Let's solve the problem step by step:
We are given two polynomials:
1. The numerator: [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]
2. The denominator: [tex]\( x^3 - 3 \)[/tex]
We need to divide the numerator by the denominator to find the quotient.
Step 1: Set up the division.
- We will use polynomial long division to divide [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
Step 2: Divide the leading terms.
- Divide the leading term of the numerator, [tex]\( x^4 \)[/tex], by the leading term of the denominator, [tex]\( x^3 \)[/tex], to get the first term of the quotient: [tex]\( x \)[/tex].
Step 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 polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]. This gives us the new polynomial:
[tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].
Step 4: Repeat the process.
- Now, divide the leading term of the new polynomial [tex]\( 5x^3 \)[/tex] by the leading term of the divisor [tex]\( x^3 \)[/tex] to get the next term of the quotient: [tex]\( +5 \)[/tex].
- Multiply the divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( 5 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from [tex]\( 5x^3 - 15 \)[/tex] to get 0.
Since there is no remainder, our division is complete, and we find that the quotient is:
[tex]\[ x + 5 \][/tex]
So, the quotient of [tex]\( (x^4 + 5x^3 - 3x - 15) \)[/tex] divided by [tex]\( (x^3 - 3) \)[/tex] is [tex]\( x + 5 \)[/tex].
We are given two polynomials:
1. The numerator: [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]
2. The denominator: [tex]\( x^3 - 3 \)[/tex]
We need to divide the numerator by the denominator to find the quotient.
Step 1: Set up the division.
- We will use polynomial long division to divide [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
Step 2: Divide the leading terms.
- Divide the leading term of the numerator, [tex]\( x^4 \)[/tex], by the leading term of the denominator, [tex]\( x^3 \)[/tex], to get the first term of the quotient: [tex]\( x \)[/tex].
Step 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 polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]. This gives us the new polynomial:
[tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].
Step 4: Repeat the process.
- Now, divide the leading term of the new polynomial [tex]\( 5x^3 \)[/tex] by the leading term of the divisor [tex]\( x^3 \)[/tex] to get the next term of the quotient: [tex]\( +5 \)[/tex].
- Multiply the divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( 5 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from [tex]\( 5x^3 - 15 \)[/tex] to get 0.
Since there is no remainder, our division is complete, and we find that the quotient is:
[tex]\[ x + 5 \][/tex]
So, the quotient of [tex]\( (x^4 + 5x^3 - 3x - 15) \)[/tex] divided by [tex]\( (x^3 - 3) \)[/tex] is [tex]\( x + 5 \)[/tex].
