Use polynomial long division to find the quotient and the remainder when 2x 3 +x 2 +3x−1 is divided by x+4 . Also, check your answer by showing that 2x 3 +x 2 +3x−1 is equal to x+4 times the quotient, plus the remainder.
B.
dividend = quotient x divisor + remainder
=________ x______ +________
To use polynomial long division to find the quotient and remainder when dividing 2x^3 + x^2 + 3x - 1 by x + 4, follow these steps:
Step 1: Write the dividend and divisor in descending powers of x.
Dividend: 2x^3 + x^2 + 3x - 1
Divisor: x + 4
Step 2: Divide the highest power term of the dividend by the highest power term of the divisor.
In this case, divide 2x^3 by x, which gives you 2x^2.
2x^2 + ____
Step 3: Multiply the entire divisor (x + 4) by the quotient you found in the previous step (2x^2), and write the result below the dividend.
(2x^2) multiplied by (x + 4) gives you 2x^3 + 8x^2.
2x^2 + ________
2x^3 + 8x^2
Step 4: Subtract the result obtained in Step 3 from the dividend.
Subtract 2x^3 + 8x^2 from 2x^3 + x^2 + 3x - 1, which gives you -7x^2 + 3x - 1.
2x^2 + ________
2x^3 + 8x^2
- ____
-7x^2 + 3x - 1
Step 5: Repeat Steps 2-4 with the remaining terms.
Divide -7x^2 by x, which gives you -7x. Write this in the next line of the quotient.
2x^2 - 7x + ____
Multiply the divisor (x + 4) by the quotient (-7x), which gives you -7x^2 - 28x.
2x^2 - 7x + ________
2x^3 + 8x^2
- ____
-7x^2 + 3x - 1
- ______
-7x^2 - 28x
Subtract -7x^2 - 28x from -7x^2 + 3x - 1, which gives you 31x - 1.
2x^2 - 7x + ________
2x^3 + 8x^2
- ____
-7x^2 + 3x - 1
- ______
-7x^2 - 28x
- ______
31x - 1
Step 6: Since the degree of the remainder (31x - 1) is less than the degree of the divisor (x + 4), we have our quotient and remainder.
The quotient is 2x^2 - 7x and the remainder is 31x - 1.
To check our answer, we can use the formula: Dividend = Quotient x Divisor + Remainder.
Let's substitute our values:
2x^3 + x^2 + 3x - 1 = (2x^2 - 7x)(x + 4) + (31x - 1)
Multiply (2x^2 - 7x)(x + 4) and simplify:
2x^3 + x^2 + 3x - 1 = 2x^3 - 7x^2 + 8x^2 - 28x + 31x - 124 + 31x - 1
= 2x^3 + x^2 + 3x - 1
The left side of the equation is equal to the right side, which confirms the correctness of our solution.