%2F(4x%2B5).jpeg "(4x^3-7x^2-11x+5)/(4x+5)")
Polynomial Long Division: (4x^3-7x^2-11x+5)/(4x+5)
This article will demonstrate how to perform polynomial long division on the expression (4x^3-7x^2-11x+5)/(4x+5).
Step 1: Setting up the Division
First, set up the division problem. The dividend (4x^3-7x^2-11x+5) goes inside the division symbol, and the divisor (4x+5) goes outside.
_______
4x+5 | 4x^3-7x^2-11x+5
Step 2: Dividing the Leading Terms
We begin by dividing the leading term of the dividend (4x^3) by the leading term of the divisor (4x). This gives us x^2. Write x^2 above the division symbol, aligning it with the x^3 term.
x^2
4x+5 | 4x^3-7x^2-11x+5
Step 3: Multiplying and Subtracting
Next, multiply the divisor (4x+5) by the term we just wrote (x^2). This gives us 4x^3 + 5x^2. Write this result below the dividend, aligning terms with the same powers of x.
x^2
4x+5 | 4x^3-7x^2-11x+5
-(4x^3+5x^2)
Now, subtract the entire expression we just wrote from the dividend. This leaves us with -12x^2 -11x.
x^2
4x+5 | 4x^3-7x^2-11x+5
-(4x^3+5x^2)
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-12x^2-11x
Step 4: Repeating the Process
Repeat the process from step 2. Divide the leading term of the new dividend (-12x^2) by the leading term of the divisor (4x), which gives us -3x. Write -3x above the division symbol, aligning it with the x^2 term.
x^2 -3x
4x+5 | 4x^3-7x^2-11x+5
-(4x^3+5x^2)
-------------
-12x^2-11x
Multiply the divisor (4x+5) by -3x, which gives us -12x^2 -15x. Write this result below the previous line, aligning terms with the same powers of x.
x^2 -3x
4x+5 | 4x^3-7x^2-11x+5
-(4x^3+5x^2)
-------------
-12x^2-11x
-(-12x^2-15x)
Now, subtract the entire expression we just wrote from the previous line. This leaves us with 4x+5.
x^2 -3x
4x+5 | 4x^3-7x^2-11x+5
-(4x^3+5x^2)
-------------
-12x^2-11x
-(-12x^2-15x)
-------------
4x+5
Step 5: Final Step
Repeat the process one more time. Divide the leading term of the new dividend (4x) by the leading term of the divisor (4x), which gives us 1. Write 1 above the division symbol, aligning it with the constant term.
x^2 -3x +1
4x+5 | 4x^3-7x^2-11x+5
-(4x^3+5x^2)
-------------
-12x^2-11x
-(-12x^2-15x)
-------------
4x+5
-(4x+5)
Multiply the divisor (4x+5) by 1, which gives us 4x+5. Write this result below the previous line.
Subtract the entire expression we just wrote from the previous line. This leaves us with 0.
x^2 -3x +1
4x+5 | 4x^3-7x^2-11x+5
-(4x^3+5x^2)
-------------
-12x^2-11x
-(-12x^2-15x)
-------------
4x+5
-(4x+5)
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0
Solution
We have reached a remainder of 0. Therefore, the solution to the division problem is:
(4x^3-7x^2-11x+5)/(4x+5) = x^2 - 3x + 1