if p(x) = 9x^2 -7 SQRT(5x) + 8 find the p(3 sqrt 5)
Answers
Answer:
x = 2/5 = 0.400
x =(-2-√8)/2=-1-√ 2 = -2.414
x =(-2+√8)/2=-1+√ 2 = 0.414
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(((5 • (x3)) + 23x2) - 9x) + 2 = 0
Step 2 :
Equation at the end of step 2 :
((5x3 + 23x2) - 9x) + 2 = 0
Step 3 :
Checking for a perfect cube :
3.1 5x3+8x2-9x+2 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 5x3+8x2-9x+2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -9x+2
Group 2: 5x3+8x2
Pull out from each group separately :
Group 1: (-9x+2) • (1) = (9x-2) • (-1)
Group 2: (5x+8) • (x2)
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(x) = 5x3+8x2-9x+2
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 5 and the Trailing Constant is 2.
The factor(s) are:
of the Leading Coefficient : 1,5
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 14.00
-1 5 -0.20 4.08
-2 1 -2.00 12.00
-2 5 -0.40 6.56
1 1 1.00 6.00
1 5 0.20 0.56
2 1 2.00 56.00
2 5 0.40 0.00 5x-2
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
5x3+8x2-9x+2
can be divided with 5x-2
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 5x3+8x2-9x+2
("Dividend")
By : 5x-2 ("Divisor")
dividend 5x3 + 8x2 - 9x + 2
- divisor * x2 5x3 - 2x2
remainder 10x2 - 9x + 2
- divisor * 2x1 10x2 - 4x
remainder - 5x + 2
- divisor * -x0 - 5x + 2
remainder 0
Quotient : x2+2x-1 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring x2+2x-1
The first term is, x2 its coefficient is 1 .
The middle term is, +2x its coefficient is 2 .
The last term, "the constant", is -1
Step-1 : Multiply the coefficient of the first term by the constant 1 • -1 = -1
Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is 2 .
-1 + 1 = 0
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 3 :
(x2 + 2x - 1) • (5x - 2) = 0
Step 4 :
Theory - Roots of a product :
4.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.