How do you divide #(x^5 - 2x^2 + 4) ÷ (x - 4)#?

Question

Elijah Allen · Accepted Answer

#(x^5-2x^2+4) div (x-4)#
#color(white)("XXXXXX")x^4+4x^3+16x^2+62x+248# Remainder: #996#

It is basically just long division, but remember to include terms with #0# value coefficients:

Chloe Alexander · Answer

undefined To divide $ (x^5 - 2x^2 + 4) $ by $ (x - 4) $, you can use polynomial long division or synthetic division. Let's use polynomial long division here.

Step 1: Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.

$$ \frac{x^5}{x} = x^4 $$

Step 2: Multiply the entire divisor $ (x - 4) $ by the term you found in step 1, then subtract this from the dividend.

$$ (x - 4)(x^4) = x^5 - 4x^4 $$

$$ (x^5 - 2x^2 + 4) - (x^5 - 4x^4) = -4x^4 - 2x^2 + 4 $$

Step 3: Repeat the process with the result from step 2.

$$ \frac{-4x^4}{x} = -4x^3 $$

$$ (x - 4)(-4x^3) = -4x^4 + 16x^3 $$

$$ (-4x^4 - 2x^2 + 4) - (-4x^4 + 16x^3) = 16x^3 - 2x^2 + 4 $$

Step 4: Repeat until the degree of the remainder is less than the degree of the divisor.

$$ \frac{16x^3}{x} = 16x^2 $$

$$ (x - 4)(16x^2) = 16x^3 - 64x^2 $$

$$ (16x^3 - 2x^2 + 4) - (16x^3 - 64x^2) = 62x^2 + 4 $$

Since the degree of the remainder (2) is less than the degree of the divisor (1), we stop.

Therefore, $ (x^5 - 2x^2 + 4) $ divided by $ (x - 4) $ equals $ x^4 - 4x^3 + 16x^2 + \frac{62x^2 + 4}{x - 4} $.

HIX Tutor · Answer

undefined To divide (x^5 - 2x^2 + 4) by (x - 4), we can use long division.

First, we look at the highest power of x in the numerator, which is x^5. We divide it by x, which gives us x^4. We write x^4 above the line.

Then, we multiply (x - 4) by x^4, which gives us x^5 - 4x^4. We subtract that from the numerator (x^5 - 2x^2 + 4).

Next, we bring down the next term, which is -2x^2. Now, we have x^4 and -2x^2 on top.

We repeat the process by dividing x^4 by x, which gives us x^3. We write x^3 above the line.

Then, we multiply (x - 4) by x^3, which gives us x^4 - 4x^3. We subtract that from the remaining terms.

We bring down the next term, which is 0x (because there is no x term).

We divide x^3 by x, which gives us another x^2. We write x^2 above the line.

We multiply (x - 4) by x^2, which gives us x^3 - 4x^2. We subtract that from the remaining terms.

Now, we bring down the next term, which is 0x again.

We divide x^2 by x, which gives us x. We write x above the line.

We multiply (x - 4) by x, which gives us x^2 - 4x. We subtract that from the remaining terms.

Lastly, we bring down the last term, which is 4.

We divide 4 by x, which gives us 0.

So, the final division is x^4 + x^3 + x^2 + x + 0 with a remainder of 4.

Therefore, (x^5 - 2x^2 + 4) ÷ (x - 4) equals x^4 + x^3 + x^2 + x with a remainder of 4.

HIX Tutor · Answer

undefined To divide (x^5 - 2x^2 + 4) ÷ (x - 4), we can use long division. Here are the steps:

Step 1: Write the dividend (x^5 - 2x^2 + 4) and the divisor (x - 4). Set up the long division like this:

_______
    x - 4 | x^5 - 2x^2 + 4

Step 2: Divide the first term of the dividend (x^5) by the first term of the divisor (x). Write the quotient (x^4) above the line.

x^4
        __________
    x - 4 | x^5 - 2x^2 + 4

Step 3: Multiply the quotient (x^4) by the divisor (x - 4) and write the result (x^5 - 4x^4) below the dividend. Subtract this result from the original dividend.

x^4
        __________
    x - 4 | x^5 - 2x^2 + 4
           - (x^5 - 4x^4)
        ______________
                  4x^4 - 2x^2 + 4

Step 4: Bring down the next term from the dividend, which is - 2x^2.

x^4
        __________
    x - 4 | x^5 - 2x^2 + 4
           - (x^5 - 4x^4)
        ______________
                  4x^4 - 2x^2 + 4
                  -  (- 2x^2)
        _________________
                  4x^4 + 0x^2 + 4

Step 5: Divide the first term of the new dividend (4x^4) by the first term of the divisor (x). Write the quotient (4x^3) above the line.

x^4 + 4x^3
        ______________
    x - 4 | x^5 - 2x^2 + 4
           - (x^5 - 4x^4)
        ______________
                  4x^4 - 2x^2 + 4
                  -  (- 2x^2)
        _________________
                  4x^4 + 0x^2 + 4
                  -(4x^4 - 16x^3)
        ______________________
                             16x^3 + 4

Step 6: Bring down the next term from the dividend, which is 4.

x^4 + 4x^3
        ______________
    x - 4 | x^5 - 2x^2 + 4
           - (x^5 - 4x^4)
        ______________
                  4x^4 - 2x^2 + 4
                  -  (- 2x^2)
        _________________
                  4x^4 + 0x^2 + 4
                  -(4x^4 - 16x^3)
        ______________________
                             16x^3 + 4
                             - (16x^3 - 64x^2)
        ________________________________
                                      64x^2 + 4

Step 7: There are no more terms to bring down. The division is now complete. The quotient is x^4 + 4x^3 and the remainder is 64x^2 + 4.

Therefore, (x^5 - 2x^2 + 4) ÷ (x - 4) = x^4 + 4x^3 with a remainder of 64x^2 + 4.