If x=5, what is the value of x3-12x2+48x-64
Answers
Answer:
STEP
1
:
Equation at the end of step 1
(((x3) - (22•3x2)) + 48x) - 64
STEP
2
:
Checking for a perfect cube
2.1 x3-12x2+48x-64 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-12x2+48x-64
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3-64
Group 2: 48x-12x2
Pull out from each group separately :
Group 1: (x3-64) • (1)
Group 2: (x-4) • (-12x)
Answer:
(x - 4)3
Step-by-step explanation:
Equation at the end of step 1
(((x3) - (22•3x2)) + 48x) - 64
STEP
2
:
Checking for a perfect cube
2.1 x3-12x2+48x-64 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-12x2+48x-64
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3-64
Group 2: 48x-12x2
Pull out from each group separately :
Group 1: (x3-64) • (1)
Group 2: (x-4) • (-12x)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
2.3 Find roots (zeroes) of : F(x) = x3-12x2+48x-64
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integersThe Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is -64.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,8 ,16 ,32 ,64
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -125.00
-2 1 -2.00 -216.00
-4 1 -4.00 -512.00
-8 1 -8.00 -1728.00
-16 1 -16.00 -8000.00
-32 1 -32.00 -46656.00
-64 1 -64.00 -314432.00
1 1 1.00 -27.00
2 1 2.00 -8.00
4 1 4.00 0.00 x-4
8 1 8.00 64.00
16 1 16.00 1728.00
32 1 32.00 21952.00
64 1 64.00 216000.00
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x3-12x2+48x-64
can be divided with x-4
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3-12x2+48x-64
("Dividend")
By : x-4 ("Divisor")
dividend x3 - 12x2 + 48x - 64
- divisor * x2 x3 - 4x2
remainder - 8x2 + 48x - 64
- divisor * -8x1 - 8x2 + 32x
remainder 16x - 64
- divisor * 16x0 16x - 64
remainder 0
Quotient : x2-8x+16 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2-8x+16
The first term is, x2 its coefficient is 1 .
The middle term is, -8x its coefficient is -8 .
The last term, "the constant", is +16
Step-1 : Multiply the coefficient of the first term by the constant 1 • 16 = 16
Step-2 : Find two factors of 16 whose sum equals the coefficient of the middle term, which is -8 .
-16 + -1 = -17
-8 + -2 = -10
-4 + -4 = -8 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and -4
x2 - 4x - 4x - 16
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-4)
Add up the last 2 terms, pulling out common factors :
4 • (x-4)
Step-5 : Add up the four terms of step 4 :
(x-4) • (x-4)
Which is the desired factorization
Multiplying Exponential Expressions:
2.6 Multiply (x-4) by (x-4)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-4) and the exponents are :
1 , as (x-4) is the same number as (x-4)1
and 1 , as (x-4) is the same number as (x-4)1
The product is therefore, (x-4)(1+1) = (x-4)2
Multiplying Exponential Expressions:
2.7 Multiply (x-4)2 by (x-4)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-4) and the exponents are :
2
and 1 , as (x-4) is the same number as (x-4)1
The product is therefore, (x-4)(2+1) = (x-4)3
Final result :
(x - 4)3