Answer:
4.1 Pull out like factors :
4x2 + 2 = 2 • (2x2 + 1)
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(x) = 2x2 + 1
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 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 1.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 3.00 -1 2 -0.50 1.50 1 1 1.00 3.00 1 2 0.50 1.50
Polynomial Roots Calculator found no rational roots
Trying to factor by splitting the middle term
4.3 Factoring 2x2 - 9x - 5
The first term is, 2x2 its coefficient is 2 .
The middle term is, -9x its coefficient is -9 .
The last term, "the constant", is -5
Step-1 : Multiply the coefficient of the first term by the constant 2 • -5 = -10
Step-2 : Find two factors of -10 whose sum equals the coefficient of the middle term, which is -9 .
-10 + 1 = -9 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -10 and 1
2x2 - 10x + 1x - 5
Step-4 : Add up the first 2 terms, pulling out like factors :
2x • (x-5)
Add up the last 2 terms, pulling out common factors :
1 • (x-5)
Step-5 : Add up the four terms of step 4 :
(2x+1) • (x-5)
Which is the desired factorization
Final result :
2 • (2x2 + 1) —————————————————— (x - 5) • (2x + 1)
.