2b^2+b-4b+8b^2-7b^3 please solve the step by step
Answers
Answer:Final result :
(b + 4) • (b - 1)2
Step-by-step explanation:STEP
1
:
Equation at the end of step 1
(((b3) + 2b2) - 7b) + 4
STEP
2
:
Checking for a perfect cube
2.1 b3+2b2-7b+4 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: b3+2b2-7b+4
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -7b+4
Group 2: b3+2b2
Pull out from each group separately :
Group 1: (-7b+4) • (1) = (7b-4) • (-1)
Group 2: (b+2) • (b2)
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(b) = b3+2b2-7b+4
Polynomial Roots Calculator is a set of methods aimed at finding values of b for which F(b)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers b which can be expressed as the quotient of two integers
The 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 4.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 12.00
-2 1 -2.00 18.00
-4 1 -4.00 0.00 b+4
1 1 1.00 0.00 b-1
2 1 2.00 6.00
4 1 4.00 72.00
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
b3+2b2-7b+4
can be divided by 2 different polynomials,including by b-1
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : b3+2b2-7b+4
("Dividend")
By : b-1 ("Divisor")
dividend b3 + 2b2 - 7b + 4
- divisor * b2 b3 - b2
remainder 3b2 - 7b + 4
- divisor * 3b1 3b2 - 3b
remainder - 4b + 4
- divisor * -4b0 - 4b + 4
remainder 0
Quotient : b2+3b-4 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring b2+3b-4
The first term is, b2 its coefficient is 1 .
The middle term is, +3b its coefficient is 3 .
The last term, "the constant", is -4
Step-1 : Multiply the coefficient of the first term by the constant 1 • -4 = -4
Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is 3 .
-4 + 1 = -3
-2 + 2 = 0
-1 + 4 = 3 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and 4
b2 - 1b + 4b - 4
Step-4 : Add up the first 2 terms, pulling out like factors :
b • (b-1)
Add up the last 2 terms, pulling out common factors :
4 • (b-1)
Step-5 : Add up the four terms of step 4 :
(b+4) • (b-1)
Which is the desired factorization
Multiplying Exponential Expressions:
2.6 Multiply (b-1) by (b-1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (b-1) and the exponents are :
1 , as (b-1) is the same number as (b-1)1
and 1 , as (b-1) is the same number as (b-1)1
The product is therefore, (b-1)(1+1) = (b-1)2
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