Question

Simplify the following polynomial and write it in standard form:



(5x2−3x+2)−(3x+5)+(4x2+10)\left(5x^2-3x+2\right)-\left(3x+5\right)+\left(4x^2+10\right)(5x
2
−3x+2)−(3x+5)+(4x
2
+10)

Answer 1

(5x
2
−3x+2)−(3x+5)+(4x
2
+10)

Answer 2

Simplify
2x2 - 3x - 5
25 - 4x2

Factoring 2x2 - 3x - 5

The first term is, 2x2 its coefficient is 2 .
The middle term is, -3x its coefficient is -3 .
The last term, "the constant", is -5

Step-1 : Multiply the coefficient of the first term by the constant 2 • -5 = -10

Step-2 : Find two factors of -10 whose sum equals the coefficient of the middle term, which is -3 .

-10 + 1 = -9
-5 + 2 = -3 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -5 and 2
2x2 - 5x + 2x - 5

Step-4 : Add up the first 2 terms, pulling out like factors :
x • (2x-5)
Add up the last 2 terms, pulling out common factors :
1 • (2x-5)
Step-5 : Add up the four terms of step 4 :
(x+1) • (2x-5)
Which is the desired factorization

Factoring: 25-4x2

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 25 is the square of 5
Check : 4 is the square of 2
Check : x2 is the square of x1

Factorization is : (5 + 2x) • (5 - 2x)
3.3 Rewrite (5-2x) as (-1) • (2x-5)

Cancel out (2x-5) which now appears on both sides of the fraction line.

Pull out like factors :

-x - 1 = -1 • (x + 1)

-x - 1
———
2x + 5