factorise x^3+5x^2+3x-9
Answers
Answer:
Step by Step Solution:
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STEP
1
:
Equation at the end of step 1
(((0 - (x3)) + 5x2) - 3x) - 9
STEP
2
:
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
-x3 + 5x2 - 3x - 9 =
-1 • (x3 - 5x2 + 3x + 9)
Checking for a perfect cube :
3.2 x3 - 5x2 + 3x + 9 is not a perfect cube
Trying to factor by pulling out :
3.3 Factoring: x3 - 5x2 + 3x + 9
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 3x + 9
Group 2: x3 - 5x2
Pull out from each group separately :
Group 1: (x + 3) • (3)
Group 2: (x - 5) • (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 :
3.4 Find roots (zeroes) of : F(x) = x3 - 5x2 + 3x + 9
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 1 and the Trailing Constant is 9.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,9
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 x + 1
-3 1 -3.00 -72.00
-9 1 -9.00 -1152.00
1 1 1.00 8.00
3 1 3.00 0.00 x - 3
9 1 9.00 360.00
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
x3 - 5x2 + 3x + 9
can be divided by 2 different polynomials,including by x - 3
Polynomial Long Division :
3.5 Polynomial Long Division
Dividing : x3 - 5x2 + 3x + 9
("Dividend")
By : x - 3 ("Divisor")
dividend x3 - 5x2 + 3x + 9
- divisor * x2 x3 - 3x2
remainder - 2x2 + 3x + 9
- divisor * -2x1 - 2x2 + 6x
remainder - 3x + 9
- divisor * -3x0 - 3x + 9
remainder 0
Quotient : x2-2x-3 Remainder: 0
Trying to factor by splitting the middle term
3.6 Factoring x2-2x-3
The first term is, x2 its coefficient is 1 .
The middle term is, -2x its coefficient is -2 .
The last term, "the constant", is -3
Step-1 : Multiply the coefficient of the first term by the constant 1 • -3 = -3
Step-2 : Find two factors of -3 whose sum equals the coefficient of the middle term, which is -2 .
-3 + 1 = -2 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -3 and 1
x2 - 3x + 1x - 3
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-3)
Add up the last 2 terms, pulling out common factors :
1 • (x-3)
Step-5 : Add up the four terms of step 4 :
(x+1) • (x-3)
Which is the desired factorization
Multiplying Exponential Expressions:
3.7 Multiply (x-3) by (x-3)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-3) and the exponents are :
1 , as (x-3) is the same number as (x-3)1
and 1 , as (x-3) is the same number as (x-3)1
The product is therefore, (x-3)(1+1) = (x-3)2
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