Use long division to divide and use the result to factor the dividend completely.
(5x2 − 3x − 14) ÷ (x − 2)
Well, I'm more of a clown than a mathematician, but I'll give it a try! Here goes the division:
5x^2 - 3x - 14
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x - 2 | 5x^2 + 0x - 3x - 14
First, let's divide 5x^2 by x, which gives us 5x. Now we multiply that by (x - 2) and subtract it from our dividend:
5x^2 - 3x - 14
__________________
x - 2 | 5x^2 + 0x - 3x - 14
-(5x^2 - 10x)
_____________
7x - 14
Next, we divide 7x by x, which gives us 7. Multiply that by (x - 2) and subtract it again:
5x^2 - 3x - 14
__________________
x - 2 | 5x^2 + 0x - 3x - 14
-(5x^2 - 10x)
_____________
7x - 14
-(7x - 14)
_____________
0
Now we have a remainder of 0, which means (x - 2) is a factor of (5x^2 − 3x − 14).
So, the fully factored form is:
(5x^2 − 3x − 14) = (x - 2)(5x + 7)
Voilà! Clowns can do long division too, apparently.
To divide (5x^2 - 3x - 14) by (x - 2) using long division, follow the steps below:
Step 1: Divide the first term of the dividend by the first term of the divisor.
In this case, divide 5x^2 by x to get 5x.
5x
Step 2: Multiply the answer obtained in Step 1 (5x) by the divisor (x - 2).
Multiply 5x by (x - 2) to get 5x^2 - 10x.
5x
__________
x - 2 | 5x^2 - 3x - 14
- (5x^2 - 10x)
Step 3: Subtract the result obtained in Step 2 from the dividend.
Subtract (5x^2 - 10x) from (5x^2 - 3x - 14) to get 7x - 14.
5x
__________
x - 2 | 5x^2 - 3x - 14
- (5x^2 - 10x)
___________
7x - 14
Step 4: Bring down the next term of the dividend.
Bring down the -14.
5x
__________
x - 2 | 5x^2 - 3x - 14
- (5x^2 - 10x)
___________
7x - 14
- (7x - 14)
Step 5: Divide the new expression by the divisor.
In this case, divide 7x - 14 by x - 2 to get 7.
5x + 7
Step 6: Multiply the answer obtained in Step 5 (7) by the divisor (x - 2).
Multiply 7 by (x - 2) to get 7x - 14.
5x + 7
__________
x - 2 | 5x^2 - 3x - 14
- (5x^2 - 10x)
___________
7x - 14
- (7x - 14)
___________
0
Step 7: Check the remainder to see if further division is possible.
Since the remainder is 0, no further division is possible.
The division result is 5x + 7, and the factored form of the dividend is (x - 2)(5x + 7).
To use long division to divide (5x^2 - 3x - 14) by (x - 2) and factor the dividend completely, follow these steps:
Step 1: Write the dividend and the divisor in long division form.
________
x - 2 | 5x^2 - 3x - 14
Step 2: Divide the first term of the dividend (5x^2) by the first term of the divisor (x) and write the result above the line as the first term of the quotient: 5x.
________
x - 2 | 5x^2 - 3x - 14
- (5x^2 - 10x)
________
7x
Step 3: Multiply the quotient (5x) by the divisor (x - 2) and write the result below the line. Subtract this result from the dividend.
________
x - 2 | 5x^2 - 3x - 14
- (5x^2 - 10x)
________
7x - 14
Step 4: Bring down the next term of the dividend ( - 14) below the line.
________
x - 2 | 5x^2 - 3x - 14
- (5x^2 - 10x)
________
7x - 14
- (7x - 14)
_______
0
Step 5: The division is complete when the remainder is zero. The quotient in this case is 5x + 7.
So, (5x^2 - 3x - 14) divided by (x - 2) is equal to 5x + 7.
Now, let's factor the dividend completely.
The original polynomial is (5x^2 - 3x - 14).
Factoring the quadratic expression 5x^2 - 3x - 14, we can write it as the product of two linear binomials:
(5x + 7)(x - 2).
So, the fully factored form of the dividend (5x^2 - 3x - 14) is (5x + 7)(x - 2).