Factor -0.25 out of 7g+3.5 .
Answers
Answer:
3.2 Factoring: -2g3+7g2+3g-6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 3g-6
Group 2: -2g3+7g2
Pull out from each group separately :
Group 1: (g-2) • (3)
Group 2: (2g-7) • (-g2)
Step-by-step explanation:
Trying to factor by pulling out :
3.2 Factoring: -2g3+7g2+3g-6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 3g-6
Group 2: -2g3+7g2
Pull out from each group separately :
Group 1: (g-2) • (3)
Group 2: (2g-7) • (-g2)
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 :
3.3 Find roots (zeroes) of : F(g) = -2g3+7g2+3g-6
Polynomial Roots Calculator is a set of methods aimed at finding values of g for which F(g)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers g 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 -2 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 g+1
-1 2 -0.50 -5.50
-2 1 -2.00 32.00
-3 1 -3.00 102.00
-3 2 -1.50 12.00
-6 1 -6.00 660.00
1 1 1.00 2.00
1 2 0.50 -3.00
2 1 2.00 12.00
3 1 3.00 12.00
3 2 1.50 7.50
6 1 6.00 -168.00
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
-2g3+7g2+3g-6
can be divided with g+1
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : -2g3+7g2+3g-6
("Dividend")
By : g+1 ("Divisor")
dividend - 2g3 + 7g2 + 3g - 6
- divisor * -2g2 - 2g3 - 2g2
remainder 9g2 + 3g - 6
- divisor * 9g1 9g2 + 9g
remainder - 6g - 6
- divisor * -6g0 - 6g - 6
remainder 0
Quotient : -2g2+9g-6 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring 2g2-9g+6
The first term is, 2g2 its coefficient is 2 .
The middle term is, -9g its coefficient is -9 .
The last term, "the constant", is +6
Step-1 : Multiply the coefficient of the first term by the constant 2 • 6 = 12
Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is -9 .
-12 + -1 = -13
-6 + -2 = -8
-4 + -3 = -7
-3 + -4 = -7
-2 + -6 = -8
-1 + -12 = -13
1 + 12 = 13
2 + 6 = 8
3 + 4 = 7
4 + 3 = 7
6 + 2 = 8
12 + 1 = 13
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(-2g2 + 9g - 6) • (g + 1)
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