(3x^2 - 21x + 30) + (x^2 - 25) factorise it
Answers
Answer:
Equation at the end of step 1
STEP
2
:
3x2 - 21x + 30
Simplify ——————————————
x2 - 25
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
3x2 - 21x + 30 = 3 • (x2 - 7x + 10)
Trying to factor by splitting the middle term
3.2 Factoring x2 - 7x + 10
The first term is, x2 its coefficient is 1 .
The middle term is, -7x its coefficient is -7 .
The last term, "the constant", is +10
Step-1 : Multiply the coefficient of the first term by the constant 1 • 10 = 10
Step-2 : Find two factors of 10 whose sum equals the coefficient of the middle term, which is -7 .
-10 + -1 = -11
-5 + -2 = -7 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -5 and -2
x2 - 5x - 2x - 10
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-5)
Add up the last 2 terms, pulling out common factors :
2 • (x-5)
Step-5 : Add up the four terms of step 4 :
(x-2) • (x-5)
Which is the desired factorization
Trying to factor as a Difference of Squares:
3.3 Factoring: x2-25
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 : x2 is the square of x1
Factorization is : (x + 5) • (x - 5)
Canceling Out :
3.4 Cancel out (x - 5) which appears on both sides of the fraction line.
Final result :
3 • (x - 2)
———————————
x + 5
Answer:
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