factorise: 3m^3 -15m + 63
Answers
Step-by-step explanation:
m2+24m+63=0
Two solutions were found :
m = -3
m = -21
Step by step solution :
Step 1 :
Trying to factor by splitting the middle term
1.1 Factoring m2+24m+63
The first term is, m2 its coefficient is 1 .
The middle term is, +24m its coefficient is 24 .
The last term, "the constant", is +63
Step-1 : Multiply the coefficient of the first term by the constant 1 • 63 = 63
Step-2 : Find two factors of 63 whose sum equals the coefficient of the middle term, which is 24 .
-63 + -1 = -64
-21 + -3 = -24
-9 + -7 = -16
-7 + -9 = -16
-3 + -21 = -24
-1 + -63 = -64
1 + 63 = 64
3 + 21 = 24 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 3 and 21
m2 + 3m + 21m + 63
Step-4 : Add up the first 2 terms, pulling out like factors :
m • (m+3)
Add up the last 2 terms, pulling out common factors :
21 • (m+3)
Step-5 : Add up the four terms of step 4 :
(m+21) • (m+3)
Which is the desired factorization
Equation at the end of step 1 :
(m + 21) • (m + 3) = 0
Step 2 :
Theory - Roots of a product :
2.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
2.2 Solve : m+21 = 0
Subtract 21 from both sides of the equation :
m = -21
Solving a Single Variable Equation :
2.3 Solve : m+3 = 0
Subtract 3 from both sides of the equation :
m = -3
Supplement : Solving Quadratic Equation Directly
Solving m2+24m+63 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula