factorize a^3+3a^2+3a-7
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a3+3a2+3a-7
Final result :
(a2 + 4a + 7) • (a - 1)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(((a3) + 3a2) + 3a) - 7
Step 2 :
Checking for a perfect cube :
2.1 a3+3a2+3a-7 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: a3+3a2+3a-7
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 3a-7
Group 2: 3a2+a3
Pull out from each group separately :
Group 1: (3a-7) • (1)
Group 2: (a+3) • (a2)
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(a) = a3+3a2+3a-7
Polynomial Roots Calculator is a set of methods aimed at finding values of a for which F(a)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers a 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 -7.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,7
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 -8.00 -7 1 -7.00 -224.00 1 1 1.00 0.00 a-1 7 1 7.00 504.00
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
a3+3a2+3a-7
can be divided with a-1
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : a3+3a2+3a-7
("Dividend")
By : a-1 ("Divisor")
dividend a3 + 3a2 + 3a - 7 - divisor * a2 a3 - a2 remainder 4a2 + 3a - 7 - divisor * 4a1 4a2 - 4a remainder 7a - 7 - divisor * 7a0 7a - 7 remainder 0
Quotient : a2+4a+7 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring a2+4a+7
The first term is, a2 its coefficient is 1 .
The middle term is, +4a its coefficient is 4 .
The last term, "the constant", is +7
Step-1 : Multiply the coefficient of the first term by the constant 1 • 7 = 7
Step-2 : Find two factors of 7 whose sum equals the coefficient of the middle term, which is 4 .
-7 + -1 = -8 -1 + -7 = -8 1 + 7 = 8 7 + 1 = 8
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(a2 + 4a + 7) • (a - 1)
Final result :
(a2 + 4a + 7) • (a - 1)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(((a3) + 3a2) + 3a) - 7
Step 2 :
Checking for a perfect cube :
2.1 a3+3a2+3a-7 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: a3+3a2+3a-7
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 3a-7
Group 2: 3a2+a3
Pull out from each group separately :
Group 1: (3a-7) • (1)
Group 2: (a+3) • (a2)
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(a) = a3+3a2+3a-7
Polynomial Roots Calculator is a set of methods aimed at finding values of a for which F(a)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers a 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 -7.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,7
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 -8.00 -7 1 -7.00 -224.00 1 1 1.00 0.00 a-1 7 1 7.00 504.00
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
a3+3a2+3a-7
can be divided with a-1
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : a3+3a2+3a-7
("Dividend")
By : a-1 ("Divisor")
dividend a3 + 3a2 + 3a - 7 - divisor * a2 a3 - a2 remainder 4a2 + 3a - 7 - divisor * 4a1 4a2 - 4a remainder 7a - 7 - divisor * 7a0 7a - 7 remainder 0
Quotient : a2+4a+7 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring a2+4a+7
The first term is, a2 its coefficient is 1 .
The middle term is, +4a its coefficient is 4 .
The last term, "the constant", is +7
Step-1 : Multiply the coefficient of the first term by the constant 1 • 7 = 7
Step-2 : Find two factors of 7 whose sum equals the coefficient of the middle term, which is 4 .
-7 + -1 = -8 -1 + -7 = -8 1 + 7 = 8 7 + 1 = 8
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(a2 + 4a + 7) • (a - 1)
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